DOUBLE COBORDISM, FLAG MANIFOLDS
AND QUANTUM DOUBLES
VICTOR M BUHSTABER AND NIGEL RAY
Abstract. Drinfeld's construction of quantum doubles is one of sev-
eral recent advances in the theory of Hopf algebras (and their actions
on rings) which may be attractively presented within the framework of
complex cobordism; these developments were pioneered by S P Novikov
and the first author. Here we extend their programme by discussing the
geometric and homotopy theoretical interpretations of the quantum dou-
ble of the Landweber-Novikov algebra, as represented by a subalgebra of
operations in double complex cobordism. We base our study on certain
families of bounded flag manifolds with double complex structure, orig-
inally introduced into cobordism theory by the second author. We give
background information on double complex cobordism, and discuss the
cell structure of the flag manifolds by analogy with the classic Schubert
decomposition, allowing us to describe their complex oriented cohomo-
logical properties (already implicit in the Schubert calculus of Bressler
and Evens). This yields a geometrical realization of the basic algebraic
structures of the dual of the Landweber-Novikov algebra, as well as its
quantum double. We work in the context of Boardman's eightfold way,
which clarifies the relationship between the quantum double and the
standard machinery of Hopf algebroids of homology cooperations.
____________
Date: 4 November 96.
Key words and phrases. Hopf algebra, Landweber-Novikov algebra, quantum doub*
*le,
complex cobordism, double cobordism, eightfold way, flag manifold, Schubert cal*
*culus.
1
2 VICTOR M BUHSTABER AND NIGEL RAY
1. Introduction
In his pioneering work [9], Drinfeld introduced the quantum double con-
struction D(H) for a Hopf algebra H. The construction was an immediate
source of interest, and Novikov proved in [19] that when H is cocommutative
then D(H) may be expressed as the smash product (in the sense of [27])
of H with its dual. Novikov further observed that when H acts appropri-
ately on any ring R, then the smash product RH may be represented as
a ring of operators on R, and he therefore referred to RH as the operator
double (or O-double), a convention we shall follow here. In consequence,
when H is cocommutative then D(H) becomes an operator double, given
by the adjoint action of H on its dual. These aspects of Hopf algebra theory
are currently under intensive study from a variety of angles, and we refer
readers to Montgomery's book [17] for a summary of background material
and a detailed survey of the state of the art.
Novikov was actually motivated by an important example from algebraic
topology, in which the algebra of cohomology operations in complex cobor-
dism theory may be constructed as an operator double by choosing H to
be the Landweber-Novikov algebra S*, and R the complex cobordism ring
U*. This viewpoint was in turn suggested by the description of S*, due to
the first author and Shokurov [6], as an algebra of differential operators on
a certain algebraic group.
Since the Landweber-Novikov algebra is cocommutative, its quantum dou-
ble is also an operator double, and the first author has used this property
in [8] to prove the remarkable fact that D(S*) may be faithfully represented
as a ring of operations in an extended version of complex cobordism, known
as double complex cobordism theory. In this sense, the algebraic and geo-
metric doubling procedures coincide. We shall therefore focus our attention
on D(S*), and refer readers to [1], [26] and [28] for comprehensive coverage
of basic information in algebraic topology. So far as we are aware, double
cobordism theories first appeared in the second author's thesis [23] and in the
associated work [25], where the double SU -cobordism ring was computed.
The above developments are especially appropriate in view of the history
of complex cobordism theory. It gained prominence in the context of stable
homotopy theory during the late 1960s, but was superseded in the 1970s
by Brown-Peterson cohomology because of the computational advantages
gained by working with a single prime at a time. Ravenel's book [22] gives
an exhaustive account of these events. Work such as [13] has recently led
to a resurgence of interest; this has been fuelled by mathematical physics,
which was, of course, the driving force behind Drinfeld's original study of
quantum groups.
Our principle aim in this work is to give detailed geometrical realizations
of the dual and the quantum double of the Landweber-Novikov algebra
incorporating the homotopy theory required for a full description of double
complex cobordism. The appropriate framework is provided by a family
of bounded flag manifolds with double U-structure. These manifolds were
originally constructed by Bott and Samelson in [4] (without reference to
flags or U-structures), but were introduced into complex cobordism theory
by the second author in [24]. We therefore allocate considerable space to
COBORDISM, FLAG MANIFOLDS, AND DOUBLES 3
a discussion of their algebraic topology, especially with respect to complex
oriented cohomology theories (of which complex cobordism is the universal
example). Part of this study is implicit in work of Bressler and Evens [5], and
we relate our treatment to their generalized Schubert calculus. An exciting
program to realize general Bott-Samelson varieties as flag manifolds with
combinatorial restrictions is currently under development by Magyar [15],
and we look forward to placing our work in his context as soon as possible.
We now describe the contents of each section.
In x2 we give an introduction to double complex cobordism from both the
manifold and homotopy theoretic points of view, since the foundations of the
construction seem never to have been properly documented. We follow the
lead of [8] by writing unreduced bordism functors as *( ) when emphasizing
their geometric origins; if these are of secondary importance, we revert to
the notation T*( ) for the corresponding reduced homology theory, where T
is the appropriate Thom spectrum.
Double complex cobordism DU*( ) is the universal example of a coho-
mology theory D*( ) equipped with two complex orientations, and we dis-
cuss this fundamental property in x3, paying particular attention to the
consequences for the D-homology and D-cohomology of complex projective
spaces, Grassmannians, and Thom complexes. These deliberations allow us
to introduce the subalgebra G* of the double complex cobordism ring DU* ,
and lay the foundations for our subsequent computations with cohomology
operations and flag manifolds.
In x4 we introduce the Landweber-Novikov algebra S* as a Hopf subal-
gebra of the algebra A*MU of all complex cobordism operations, and discuss
the identification of the dual of each (over Z and U* respectively) with
G* and DU* . Following Novikov, we describe A*MU as the operator double
of U* and S*. We then define the algebra A*DU of operations in double
complex cobordism by analogy, and explain the appearance of a subalge-
bra isomorphic to the quantum double D(S*). We couch our exposition
in terms of Boardman's eightfold way [2], which we believe to be the most
comprehensive framework for the multitude of actions and coactions which
arise.
We define our family of bounded flag manifolds B(Zn+1 ) in x5, and study
their geometry and topology. We describe a poset of subvarieties XQ which
serve to desingularize their cell structures, and which are closely related to
the Schubert calculus of [5]. We also introduce the basic U- and double U-
structures on the XQ which lie at the heart of our subsequent calculations,
and lead to a geometrical realization of G*.
We apply this material in x6 by computing the E-homology and cohomol-
ogy of the XQ for any complex oriented spectrum E; the methods readily
extend to doubly complex oriented spectra D, and so enable us to specify
the DU-theory normal characteristic numbers. We interpret the results in
terms of our calculus of subvarieties, deducing that G* is closed under the
action of the operator subalgebra S* S*. This leads to our description of
many of the algebraic structures in S* and S*, including the commutation
law for D(S*) considered as an operator double.
4 VICTOR M BUHSTABER AND NIGEL RAY
We shall use the following notation and conventions in later sections, and
without further comment.
We systematically confuse a complex vector bundle ae with its classifying
map into the appropriate Grassmannian, and write C m for the trivial com-
plex m-plane bundle over any space X. We denote the universal complex
m-plane bundle over BU (m) by (m), so that (1) is the Hopf line bundle
over complex projective space CP 1 ; we often abbreviate (1) to , especially
over a finite skeleton CP m . If ae is a complex m-plane bundle whose base
is a finite CW complex, we let ae? denote the complementary (p - m)-plane
bundle in some suitably high dimensional trivial bundle C p.
We write . for the space consisting of a single point, and X+ for its
disjoint union with an arbitrary space X.
The Hopf algebras we use are intrinsically geometrical and naturally
graded by dimension, as are ground rings such as U*. Sometimes our alge-
bras are not of finite type, and must therefore be topologized when form-
ing duals and tensor products; this has little practical effect, but is fully
explained in [3], for example. Duals are invariably taken in the graded
sense and we adapt our notation accordingly. Thus we write A*MU for
the algebra of complex cobordism operations, and AMU* for its continuous
dual Hom U* (A*MU; U*), which in turn forces us to write S* for the graded
Landweber-Novikov algebra, and S* for its dual Hom Z(S*; Z); neither of
these notations is entirely standard.
Several of our algebras are polynomial in variables such as bk of grading 2k,
where b0 is the identity. An additive basis is therefore given by monomials
of the form b!11b!22: :b:!nn, which we denote by b!, where ! is the sequence
of nonnegative, eventually zero integers (!1; !2; : :;:!n; 0 : :):. The set of *
*all
such sequences forms anPadditive semigroup, and b b! = b +! . Given any
!, we write |!| for 2 i!i, which is the grading of b!. We distinguish the
sequences ffl(k), which have a single nonzero element 1 and are defined by
bffl(k)= bkPfor each integer k 1. It is often convenient to abbreviate the
formal sum k0 bk to b, in which case we write (b)nkfor the component of
the nth power of b in grading 2k; negative values of n are permissible.
When dualizing, we choose dual basis elements of the form c!, defined by
= ffi!; ; this notation is designed to be consistent with our conventi*
*on
on gradings, and to emphasize that the elements c! are not necessarily
monomials themselves.
Unless otherwise indicated, tensor products are taken over Z.
2. Double complex cobordism
In this section we outline the theory of double complex cobordism, con-
sidering both the manifold and homotopy theoretic viewpoints.
Like all cobordism theories, double complex cobordism is based on a class
of manifolds whose stable normal bundle admits a specific structure. Once
this structure is made precise, then the standard procedures described by
Stong [26] may be invoked to construct the bordism and cobordism functors
on a suitable category of topological spaces, and to describe them from the
COBORDISM, FLAG MANIFOLDS, AND DOUBLES 5
homotopy theoretic viewpoint in terms of the corresponding Thom spec-
trum. Nevertheless, in view of the fact that more general indexing sets are
required than those considered by Stong, we explain some of the details.
Philosophically, double complex cobordism theory is based on manifolds
M whose stable normal bundle M (abbreviated to whenever the context
allows) possesses a specific splitting ~=` r into two complex bundles,
which we often label the left and right components.
More precisely, given any positive integers m and n let us write U(m; n)
for the product of unitary groups U(m) x U(n), so that the classifying
space BU (m; n) may be canonically identified with BU (m) x BU (n). Thus
BU (m; n) carries the complex (m + n)-plane bundle (m; n), defined as
(m)x(n) and classified by the Whitney sum map BU (m; n) ! BU (m+n).
The standard inclusion of U(m) in the orthogonal group O(2m) induces a
map of classifying spaces fm;n :BU (m; n) ! BO (2(m + n)), and the stan-
dard inclusion of U(m) in U(m + 1) induces a map of classifying spaces
gm;n :BU (m; n) ! BU (m + 1; n + 1). These maps constitute a doubly in-
dexed version of a (B; f) structure in the sense of Stong, although care is
required to ensure that they are sufficiently compatible over both m and n.
There are also product maps
(2.1) BU (m; n) x BU (p; q) -! BU (m + p; n + q)
induced by Whitney sum, whose compatibility is more subtle, but confirms
that the corresponding (B; f) cobordism theory is multiplicative; this is our
double complex cobordism theory, referred to in [23] as U x U theory.
We therefore define a double U-structure on M to consist of an equivalence
class of lifts of to BU (m; n), for some values of m and n which are suitably
large compared with the dimension of M. This class of lifts provides the
isomorphism ~=` r, where ` and r are classified by the left and right
projections onto the respective factors BU (m) and BU (n). If we wish to
record a particular choice of m and n, we may refer to the resulting U(m; n)-
structure. Given such a structure on M, it is convenient to write O(M) for
M invested with the U(n; m)-structure induced by the obvious switch map
BU (m; n) ! BU (n; m); we emphasise that M and O(M) are, in general,
distinct. Any manifold with a U(m; n)-structure has a U(m + n)-structure,
obtained by forgetting the splitting. If M has a U(m; n)-structure and N has
a U(p; q)-structure, then the product U(m + p; n + q) structure on M x N is
given by choosing MxN` and MxNr to be M`xN` and MrxNr respectively.
A typical example, of which we shall use analogues in Theorem 6.8 and its
applications, is provided by complex projective space CP n-1, whose stable
normal bundle is isomorphic to -n. If we select ` and r to be -k and
(k-n) respectively, we obtain infinitely many distinct double U-structures.
Also, if M and N admit U-structures, then M xN admits the product double
U-structure.
We may choose to impose all our structures on the stable tangent bundle
oM of M, so long as we observe the usual caution in choosing a canoni-
cal trivialization of o. To avoid this issue and be consistent with the
homotopy theoretic approach, we prefer to use normal structures wherever
possible.
6 VICTOR M BUHSTABER AND NIGEL RAY
The compatibility required of the maps (2.1) is most readily expressed
in the language of May's coordinate-free functors (as described, for exam-
ple, in [10]), which relies on an initial choice of infinite dimensional inner
product space Z1 , known as a universe. We may here assume that Z1 is
complex. This language was originally developed to prove that the multi-
plicative structure of complex cobordism is highly homotopy coherent [16],
and its usage establishes that the same is true for double complex cobor-
dism so long as we consistently embed our double U-manifolds in finite
dimensional subspaces V W of the universe Z1 Z1 . We define the clas-
sifying space B(V; W ) by appropriately topologizing the set of all subspaces
of V W which are similarly split. If V and W are spanned respectively
by (necessarily disjoint) m and n element subsets of some predetermined or-
thonormal basis for Z1 Z1 , we refer to them as coordinatized, and write
the classifying space as BU (m; n) to conform with our earlier notation. We
then interpret (2.1)as a coordinatized version of the Whitney sum map, on
the understanding that the subspaces of dimension m and p are orthogonal
in Z1 , as are those of dimension n and q. The Grassmannian geometry of
the universe immediately guarantees the required compatibility.
In our work below, we may safely confine such considerations to occa-
sional remarks, although they are especially pertinent when we define the
corresponding Thom spectrum and its multiplicative properties.
The double complex cobordism ring DU* consists of cobordism classes of
double U-manifolds, with the product induced as above; as we shall see,
it is exceedingly rich algebraically. The double complex bordism functor
DU* ( ) is an unreduced homology theory, defined on an arbitrary topo-
logical space X by means of bordism classes of maps into X of manifolds
with the appropriate structure; it admits a canonical involution (also de-
noted by O), induced by switching the factors of the normal bundle. Thus
DU* (X) is always a module over DU* , which in this context becomes iden-
tified with DU* (.), and is known as the coefficient ring of the theory. More-
over, the product structure ensures that DU* (X) is both a left and a right
U* -module.
Double complex cobordism *DU( ) is the dual cohomology functor, which
we define geometrically using Quillen's techniques [21]. For any double U-
manifold X, a cobordism class in *DU (X) is represented by an equivalence
class of compositions
M -i!E` Er ss-!X;
where ss is the projection of a complex vector bundle split into left and right
components, and i is an embedding of a double U-manifold M whose normal
bundle is split compatibly.
If we ignore the given splitting of each normal bundle (and simultaneously
identify Z1 Z1 isometrically with Z1 ), we obtain a forgetful homomor-
phism ss :DU* (X) ! U* (X) for any space X. Conversely, if we interpret a
given U(m)-structure as either a U(m) x U(0)-structure or a U(0) x U(m)-
structure, we obtain left and right inclusions ` and r: U* (X) ! DU* (X),
which are interchanged by O. We note that ss is an epimorphism, and that
` and r are monomorphisms because both ss O ` and ss O r are the identity.
We shall be especially interested in the action of these homomorphisms on
COBORDISM, FLAG MANIFOLDS, AND DOUBLES 7
the coefficient rings. We note further that, for any space X, the standard
product structure in *U(X) factorizes as
*U(X) *U(X) -`r--!*DU(X) *DU(X)
(2.2)
- ! *DU(X) -ss-!*U(X);
where the central homomorphism is the product in *DU(X).
The homotopy theoretic viewpoint of these functors is based on the cor-
responding Thom spectrum DU, to which we now turn.
In order to allow spectra which consist of a doubly indexed direct system
(rather than the more traditional sequence) of spaces and maps, as well
as to ensure that such spectra admit a product which is highly homotopy
coherent, it is most elegant to return to the coordinate-free setting. We
define the Thom space M(V; W ) by the standard construction on B(V; W ),
and again allow the Grassmannian geometry of the universe to provide the
necessary compatibility for both the structure and the product maps. As in
(2.1) we give explicit formulae only for coordinatized subspaces.
We write MU(m; n) for Thom complex of the bundle (m; n), which may
of course be canonically identified with MU (m) ^ MU (n). Then the coor-
dinatized structure maps take the form
(2.3) S2(p+q)^ MU (m; n) -! MU (m + p; n + q);
by Thom complexifying the classifying maps of (C pxC q) (m; n). Hence-
forth we take this direct system as our definition of the DU spectrum, noting
that the Thom complexifications of the maps (2.1)provide a product map
DU , which is highly coherent, and equipped with a unit by (2.3)in the case
m = n = 0. It is a left and right module spectrum over MU by virtue of
the systems of maps
MU (p) ^ MU (m; n)! MU (m + p; n) and
MU (m; n) ^ MU (q) ! MU (m; n + q);
which are also highly coherent by appeal to the coordinate-free setting.
This setting also enables us to define smash products of spectra [10],
and therefore to write DU as MU ^ MU . The involution O is then induced
by interchanging factors, and we may represent the bimodule structure by
maps MU ^ DU ! DU and DU ^ MU ! DU, induced by applying the
MU product MU on the left and right copies of MU ^ MU respectively.
We define the reduced bordism and cobordism functors on a topological
space X by means of
DUk(X) = lim-!m;nss2(n+m)+k(MU(m; n) ^ X)
(2.4)
and DUk(X) = lim-!m;n{S2(n+m)-k ^ X; MU (m; n)};
where the brackets { } denote based homotopy classes of maps. The graded
groups DU*(X) and DU*(X) consist of the appropriate direct sums over
k. These definitions exhibit DU*(X) as a commutative graded ring, by
virtue of the product structure on DU. The standard complex bordism and
cobordism functors are defined similarly, employing the spectrum MU in
place of DU.
8 VICTOR M BUHSTABER AND NIGEL RAY
We abbreviate lim-!m;nss2(n+m)+k(MU (m; n)) ^ X) to ssk(DU ^ X) and
lim-!m;n{S2(n+m)-k ^ X; MU (m; n)} to [S-k X; DU ] respectively; this is stan-
dard notation for stabilized groups of homotopy classes. Thus the coefficient
ring DU*(S0), written DU* for convenience, is simply the homotopy ring
ss*(DU) of DU. In similar vein, given a second homology theory E*( ) we
may introduce the E homology and cohomology groups of DU by means of
Ek(DU) = lim-!m;nE2(n+m)+k(MU (m; n))
(2.5)
and Ek(DU) = lim-m;nE2(n+m)+k (MU (m; n))
for all integers k.
Following usual practice, we define the unreduced bordism and cobordism
functors of X to be DUk(X+ ) and DUk(X+ ) respectively; the reduced and
the unreduced theories differ only by a copy of the coefficient ring, according
to the equations
DUk(X+ ) = DUk(X) DUk and DUk(X+ ) = DUk(X) DU-k ;
which arise by considering X+ as the one point union X _ S0. In this
context, we often write 1 for the element in DU*(X+ ) or DU*(X+ ) which
corresponds to the appropriate generator of DU0.
The Whitney sum map BU (m; n) ! BU (m + n) induces a forgetful map
of ring spectra ss :DU ! MU , whilst the respective inclusions of BU (m)
and BU (n) in BU (m + n) induce the inclusions ` and r: MU ! DU.
All three maps may be extended to the coordinate-free setting, and both `
and r yield the identity map after composition with ss. Moreover, the MU
product MU factorizes as
MU (m) ^ S2n^ S2p ^ MU (q) -`^r--!MU (m; n) ^ MU (p; q)
DU---!MU (m + n; p + q) -ss!MU (m + n + p + q);
in concert with (2.2). We deduce that `, r and ss all define multiplicative
transformations between the appropriate functors, and that ` and r are
interchanged by O. Given an element of MU *(X) or MU *(X), we shall
often write `() and r() as ` and r respectively.
By way of example, we may combine (2.4)and (2.5)to obtain
(2.6) DU* ~=lim-!mss2m+* (MU (m) ^ MU ) = MU *(MU );
this follows at once if we write DU as MU ^ MU . In fact MU *(MU ) is
the Hopf algebroid of cooperations in MU homology theory, and we shall
discuss it in considerable detail in x4 below. Suffice it to say here that its
associated homological algebra has been extensively studied in connection
with the Adams-Novikov spectral sequence and the stable homotopy groups
of spheres. For detailed calculations, however, it has proven more efficient
to concentrate on a single prime p at a time, and work with the p -local
summand BP*(BP ) given by Brown-Peterson homology [22].
There is a natural isomorphism between the manifold and the homotopy
theoretic versions of any bordism functor. This stems from the Pontryagin-
Thom construction, which we may summarize in the case of the coefficient
ring for double complex cobordism as follows. Consider any manifold Mk
embedded in Sk+2(m+n) with a U(m; n)-structure, and collapse to 1 the
COBORDISM, FLAG MANIFOLDS, AND DOUBLES 9
complement of a normal neighbourhood. Composing with the Thom com-
plexification of the classifying map for yields a map from Sk+2(m+n) to
MU (m; n), and thence a homomorphism DUk ! ssk(DU). This defines the
promised isomorphism DU* ~=DU*, although the verification that it has
the necessary algebraic properties requires considerable work, and depends
upon Thom's transversality theorems. We note that the isomorphism maps
the geometric involution given by interchanging the factors of the normal
bundle to the homotopy theoretic involution given by interchanging the fac-
tors of the DU spectrum, and that it may, with further care, be naturally
extended to the coordinate-free setting. Henceforth, we shall pass regu-
larly between the manifold and homotopy theoretic viewpoints, assuming
the Pontryagin-Thom construction wherever necessary.
3. Orientation classes
In this section we explain how DU is the universal example of a spectrum
admitting two distinct complex orientations, and consider the consequences
for the double complex bordism and cobordism groups of some well-known
spaces in complex geometry. We first recall certain basic definitions and
results, which may be found, for example in [1].
We assume throughout that E is a commutative ring spectrum, whose
zeroth homotopy group E0 is isomorphic to the integers. Then E is complex
oriented if the cohomology group E2(CP 1 ) contains an element xE , known
as the orientation class, whose restriction to E2(CP 1) is a generator when
the latter group is identified with E0. Under these circumstances, we may
deduce that the free E*-module E*(CP 1 ) consists of formal power series
in xE , whose powers define dual basis elements fiEk in E2k(CP 1 ). If we
continue to write fiEk for its image under the inclusion of BU (1) in BU (m)
(for any value of m, including 1), then E*(BU (m)) is the free E*-module
generated by commutative monomials of length at most m in the elements
fiEk. For 1 k m, the duals of the powers of fiE1 define the Chern classes
cEk in E2k(BU (m)), which generate E*(BU (m)) as a polynomial algebra
over E*; clearly cE1agrees with xE over CP 1 . When we pass to the direct
limit over m, we obtain
(3.1) E*(BU ) ~=E*[fiEk: k 0] and E*(BU ) ~=E*[[cEk: k 0]];
where the Pontryagin product in homology is induced by Whitney sum. We
write monomial basis elements in the fiEk as (fiE )! for any sequence !, and
their duals as cE!. Thus cE(k)and cEkcoincide.
Considering BU (m - 1) as a subspace of BU (m), we may express the
Thom complex MU (m) of (m) as the quotient space BU (m)=BU (m - 1),
at least up to homotopy equivalence and for any finite m. Thus E*(MU (m))
and E*(MU (m)) may be computed from the cofiber sequence
(3.2) BU (m - 1) -! BU (m) -! MU (m)
by applying E homology and cohomology respectively; the resulting two
sequences of E*-modules are short exact. We may best express the con-
sequences in terms of Thom isomorphisms, for which we first identify the
pullback of cEmin E2m (MU (m)) as the Thom class tE (m) of (m), observing
10 VICTOR M BUHSTABER AND NIGEL RAY
that its restriction over the base point is a generator of E2m (S2m ) when the
latter is identified with E0. Indeed, this property reduces to the defining
property for xE when m is 1 (thereby identifying xE as the Thom class of
(1)), and the same generator arises for all values of m. It follows from the
definitions that tE (m + n) pulls back to the external product tE (m)tE (n) in
E2(m+n)(MU (m) ^ MU (n)) under MU . The homomorphisms
OE*: Ek+2m (MU (m)) ! Ek(BU (m)+ )
(3.3)
and OE*: Ek(BU (m)+ ) ! Ek+2m (MU (m));
determined by the relative cap and cup products with tE (m), are readily
seen to be isomorphisms of E*-modules for all integers k 0; they are
known as the E theory Thom isomorphisms for (m). We define elements
bEkin E2(k+m)(MU (m)) as OE-1*(fik), and elements sEkin E2(k+m)(MU (m))
as OE*(cEk); each of these families extends to a set of generators over E* in
the appropriate sense.
We may stabilize the Thom isomorphisms by allowing m to become infi-
nite, in which case (3.1)yields the descriptions
(3.4) E*(MU ) ~=E*[bEk: k 0] and E*(MU ) ~=E*[[sEk: k 0]];
where bEklies in E2k(MU ) and sEkin E2k(MU ), for all k 0. We emphasise
that the multiplicative structure in homology is induced by MU , but that
in cohomology it exists only as an algebraic consequence of OE*, and is not
given by any cup product. We continue to write monomial basis elements
in the bEkas (bE )!, and their duals as sE!, for any sequence !. Again, sE(k)
and sEkcoincide. We write tE in E0(MU ) for the stable Thom class, which
corresponds to the element 1 in the description (3.4), and is represented by
a multiplicative map of ring spectra.
We have therefore described a procedure for constructing tE from our
initial choice of xE ; in fact this provides a bijection between complex orien-
tation classes in E and multiplicative maps MU ! E.
When m is 1, the cofiber sequence (3.2)reduces to the homotopy equiv-
alence CP 1 ! MU (1), and fiEk is identified with bEk-1under the map in-
duced in E homology. The Thom isomorphisms satisfy OE*(bEk) = fiEk-1and
OE*((xE )k-1) = (xE )k respectively, for all k 1.
Any complex m -plane bundle ae over a space X has a Thom class tE (ae)
in E2m (M(ae)), obtained by pulling back the universal example tEmalong the
classifying map for ae. We may use this Thom class exactly as in (3.3)to
define Thom isomorphisms
OE*: Ek+2m (M(ae)) ! Ek(X+ ) and OE*: Ek(X+ ) ! Ek+2m (M(ae)):
If ae is a virtual bundle its Thom space is stable, and so long as we insist
that its bottom cell has dimension zero, we acquire Thom isomorphisms in
the format of (3.4).
We remark that MU is itself complex oriented if we choose xMU to be
represented by the homotopy equivalence CP 1 ! MU (1); we shall abbrevi-
ate this class to x. The resulting Thom class t is represented by the identity
map on MU . In fact MU is the universal example, since any Thom class
tE induces a homomorphism MU * ! E* in homotopy, which extends to
COBORDISM, FLAG MANIFOLDS, AND DOUBLES 11
the unique homomorphism MU *(CP 1 ) ! E*(CP 1 ) satisfying xMU 7! xE .
In view of these properties, we shall dispense with the superscript MU in
the universal case wherever possible. So far as Quillen's geometrical inter-
pretation of cobordism is concerned, a Thom class t(ae) in 2mU(M(ae)) is
represented by the inclusion of the zero section M M(ae), whenever ae lies
over a U-manifold M.
We combine (2.6)with the Thom isomorphism MU *(MU ) ~=MU *(BU +)
to obtain a left MU *-isomorphism h: DU* ~= MU *(BU +), which has an
important geometrical interpretation.
Proposition 3.5. Suppose that an element of DU* is represented by a man-
ifold Mk with double U-structure ` r; then its image under h is repre-
sented by the singular U-manifold r: Mk ! BU .
Proof. By definition, the image we seek is represented by the composition
Sk+2(m+n) ! M() ! MU (m) ^ MU (n)
-1^ffi-!MU(p) ^ MU (m) ^ BU (m) ^1
+ - -! MU (p + m) ^ BU (m)+ ;
where the first map is obtained by applying the Pontryagin-Thom construc-
tion to an appropriate embedding Mk Sk+2(m+n) , and the second classifies
the double U-structure on Mk. We may identify the final three maps as the
Thom complexification of the composition
Mk -`r--! BU (m) x BU (n) -1xffi-!BU(m)x BU (n) x BU (n)
-x1--!BU (m + n) x BU (n);
__
which simplifies to r, as sought. |__|
Corollary 3.6. Suppose that an element of U* (BU +) is represented by a
singular U-manifold f :Mk ! BU (q) for suitably large q; then its inverse
image under h is represented by the double U-structure ( f? ) f on M.
We remark that our proof of Proposition 3.5 shows that h is actually mul-
tiplicative, so long as we invest MU *(BU +) with the Pontryagin product
which arises from the Whitney sum map on BU . Moreover, h conjugates
the involution O so as to act on MU *(BU +), where it interchanges the map
f of Corollary 3.6 with f? .
Returning to our complex oriented spectrum E, we record the fundamen-
tal relationship with the theory of formal groups, as introduced by Novikov
[18] and developed by Quillen in his celebrated work [20]. It depends on the
fact that the K"unneth isomorphism identifies E*(CP 1 xCP 1) with the ring
of formal power series E*[[xE ; yE ]], where xE and yE denote the pullbacks
of xE from projection onto the first and second factors CP 1 respectively.
Since the first Chern class cE1( ) of the external tensor product lies in
E2(CP 1 x CP 1 ), we obtain a formal power series expansion
X
(3.7) cE1( ) = xE + yE + aEi;j(xE )i(yE )j;
i;j1
where each aEi;jlies in the coefficient group E2(i+j-1). This formal power
series, which we usually denote by F E(X; Y ), is the formal group law for
E. It is associative, commutative, and 1-dimensional, and admits an inverse
12 VICTOR M BUHSTABER AND NIGEL RAY
induced by complex conjugation on CP 1 . The universal example is provided
by the spectrum MU , in which case the coefficients MU *form the Lazard
ring L; it is well-known that L is generated (as a ring, but with redundancy)
by the elements ai;j. Other examples are given by the integral Eilenberg-
MacLane spectrum H, and the complex K-theory spectrum K, for which
canonical orientations may be chosen such that
F H(X + Y ) = X + Y and F K(X + Y ) = X + Y + uXY
are the additive and multiplicative formal group laws respectively (where u
is the Bott periodicity element generating the coefficient group ss2(K)).
So long as E* is free of additive torsion, we may construct a formal power
series expE(X) which gives a strict isomorphism between the additive formal
group law and F E; this is defined over EQ* = E*Q, and was shown by the
first author [7] to be the image of xE under the Chern-Dold character (or
rationalization map). It is known as the exponential series for F E(X; Y ),
and its defining property may be expressed as
expE(X + Y ) = F E(exp E(X); expE(Y )):
Its substitutional inverse logE(X) is the logarithm for F E(X; Y ).
We refer readers to Hazewinkel's encyclopaedic book [11] for further de-
tails on formal group laws.
It is certainly true that DU is complex oriented, because we have already
defined two multiplicative maps ` and r: MU ! DU which therefore serve
as Thom classes t` and tr. The corresponding orientation classes are `(x)
and r(x) in DU2(CP 1 ), which we shall denote, as promised, by x` and xr
respectively. Thus
DU*(CP+1) ~=DU*[[x`]] ~=DU*[[xr]];
and there are inverse formal power series
X X __
xr = gkxk+1` and x` = gkxk+1r;
k0 k0
written g(x`) and __g(xr) respectively. Clearly the elements gk and __gklie in
DU2k for all k, and are interchanged by the involution O. Furthermore,
g0 = __g0= 1 because x` and xr restrict to the same generator of DU2(S2).
The geometrical significance of the gk is illustrated by the following property.
Proposition 3.8. Under the isomorphism h, we have that
h(gk) = fik
in MU2k(BU +), for all k 0.
Proof. We may express gk as the Kronecker product in DU2k,
which is represented by the composition
fik+1 1 1^x
S2(p+k+1)---! MU (p) ^ CP - -! MU (p) ^ MU (1);
for suitably large p. This stabilizes to bk in MU2k(MU ), and hence to_fik_in
MU2k(BU +), as required. |__|
Corollary 3.9. The subalgebra G* of DU *generated by the elements gk is
polynomial over Z.
COBORDISM, FLAG MANIFOLDS, AND DOUBLES 13
Proof. This result follows from the multiplicativity of h and the indepen-_
dence of monomials in the fik over Z. |__|
We also infer from Proposition 3.8 that gk may be realized geometrically
by choosing a singular U-manifold representing fik in MU 2k(CP 1 ), and
amending its double U-structure according to Corollary 3.6. Interchanging
` and r yields __gk. We make extensive use of the algebra G* in later sections.
We refer to any spectrum D equipped with two complex orientations
which restrict to the same element of D0 as doubly complex oriented. Ob-
viously E ^ F is such a spectrum whenever E and F are complex oriented,
but not all examples take this form. Nonetheless, we shall write the two
orientation classes as xD` and xDr respectively, so that they are related by
inverse formal power series
X X __
(3.10) xDr= gDk(xD`)k+1 and xD`= gDk(xDr)k+1
k0 k0
in D2(CP 1 ).
By mimicking the programme laid out above for E, we may construct left
and right sets of D*-generators for D*(CP 1 ), D*(BU (m)), D*(MU (m)),
and for their cohomological counterparts. Thus, for example, there are left
and right Chern classes cDk;`and cDk;rin D2k(BU (m)) for k m, and left and
right Thom classes tD`(m) and tDr(m) in D2m (MU (m)); the latter give rise
to left and right Thom isomorphisms associated to an arbitrary complex
bundle ae. There are also left and right formal group laws F`D(X; Y ) and
FrD(X; Y ).
In the light of (3.10), we would expect any left and right constructions
of this form to be interrelated by the elements gDk. This is indeed the case,
and is further testimony to their importance.
Lemma 3.11. We have that
(1)in D0(MU ),
X
tDr= tD`+ (gD )!sD!;`;
!
(2)the formal power series gD (X) provides a strict isomorphism between
the formal group laws F`D(X; Y ) and FrD(X; Y );
(3)in D2n(CP 1 ),
Xn
fiDn;`= (gD )kn-kfiDk;r
k=0
for all n 0.
Proof. For (1), we dualize (3.10) and pass to D*(BU ), then dualize back
again to D*(BU ) and apply the Thom isomorphism.
For (2), we apply (3.10)twice to obtain
F`D(gD (xDr); gD (yDr)) = F`D(xD`; yD`) and gD (FrD(xDr; yDr)) = cD1;`( )
in D2(CP 1 x CP 1 ). Thus F`D(gD (xDr); gD (yDr)) = gD (FrD(xDr; yDr)), from
(3.7).
14 VICTOR M BUHSTABER AND NIGEL RAY
P
For (3), we write (xDr)k as (xD`)k( i0 gi(xD`)i)k, and obtain
<(xDr)k; fiDn;`> = (gD )kn-k
__
as required. |__|
When DU is equipped with the orientation classes x` and xr it becomes the
universal example of a doubly complex oriented spectrum, since the exterior
product tD`tDris represented by a multiplicative spectrum map tD :DU ! D
whose induced transformation DU2(CP 1 ) ! D2(CP 1 ) maps x` and xr to
xD` and xDr respectively. It therefore often suffices to consider the case DU
(as we might in Lemma 3.11, for example). We shall continue to omit the
superscript DU whenever the context makes clear that we are dealing with
the universal case. We note from the definitions that the homomorphism of
coefficient rings DU* ! D* induced by tD satisfies
(3.12) gk 7! gDk and __gk7! __gDk
for all k 0. Thus DU* is universal for rings equipped with two formal
group laws which are linked by a strict isomorphism.
Whenever a complex vector bundle is given with a prescribed splitting
ae ~=ae` aer, then tD (ae) acts as a canonical Thom class tD`(ae`)tDr(aer), and*
* so
defines a Thom isomorphism which respects the splitting. In the universal
case, t(ae) is represented geometrically by the inclusion of the zero section
M M(ae` aer) whenever ae lies over a double U-manifold M.
As an example, it is instructive to consider the case when D is MU ,
doubly oriented by setting xD`= xDr= x. The associated Thom class is the
forgetful map ss :DU ! MU, since ss(x`) = ss(xr) = x; we therefore deduce
from (3.12)that
ss(gk) = ss(__gk) = 0
for all k > 0.
We briefly consider the D*-modules D*(BU (m; n)) and D*(MU (m; n)),
together with their cohomological counterparts. These may all be described
by application of the K"unneth formula. For example D*(BU (m; n)) is a
power series algebra, generated by any one of the four possible sets of Chern
classes
{cDj;` 1; 1 cDk;r};{cDj;` 1; 1 cDk;`};
(3.13)
{cDj;r 1; 1 cDk;`}; or {cDj;r 1; 1 cDk;r};
where 1 j m and 1 k n. The first of these are by far the most
natural, and we shall choose them whenever possible. The stable versions,
in which we take limits over one or both of m and n, are obtained by the
obvious relaxation on the range of j and k. We shall use notation such as
(fiD ) (fiD )! (omitting the subscripts ` and r) to indicate our first-choice
basis monomials in D*(BU (m; n)), and (bD ) (bD )! for their images in
D*(MU (m; n)) under the Thom isomorphism induced by tD (m; n); we write
cD cD!and sD sD!respectively for the corresponding dual basis elements
in D*(BU (m; n)) and D*(MU (m; n)). As before, it often suffices to consider
the universal example DU.
COBORDISM, FLAG MANIFOLDS, AND DOUBLES 15
4. Operations and cooperations
In this section we consider the operations and cooperations associated
with MU and DU. We work initially with our arbitrary spectra E and D,
specializing to MU and DU as required; we study both the geometric and
the homotopy theoretic aspects, using Boardman's eightfold way [2] (and its
update [3]) as a convenient algebraic framework. All comments concerning
the singly oriented E apply equally well to D unless otherwise stated.
For any integer n, the cohomology group En(E) consists of homotopy
classes of spectrum maps E ! Sn ^ E, and therefore encodes E-theory co-
homology operations of degree n. Thus E*(E) is a noncommutative, graded
E*-algebra with respect to composition of maps, and realizes the algebra A*E
of stable E-cohomology operations. It is important to observe that E*(E)
is actually a bimodule over the coefficients E*, which act naturally on the
left (as used implicitly above), but also on the argument and therefore on
the right. The same remarks apply to E*(E), on which the product map
E induces a commutative E*-algebra structure; the two module structures
are then defined respectively by the left and right inclusions j` and jr of
the coefficients in E*(E) ~=ss*(E ^ E). We shall normally maintain the con-
vention of assuming the left action without comment, and using the right
action only when explicitly stated. We refer to E*(E) as the algebra AE*of
stable E-homology cooperations, for reasons which will become clear below.
In fact E*(X) is free and of finite type for all spaces and spectra X that
we consider here. This simplifies the topologizing of E*(X), which involves
little more than accommodating the appearance of formal power series in
certain computations. Moreover, it ensures that E induces a cocommuta-
tive coproduct ffiE :E*(E) ! E*(E)b E*E*(E), where the tensor product is
completed whenever E*(E) fails to be of finite type.
We consider the E*-algebra map tE*:E*(MU) ! E*(E) induced by the
Thom class tE , and define monomials e! as tE*(bE )!. When E is singly
oriented we assume that these monomials form a basis for E*(E), which is
therefore isomorphic to the polynomial algebra E*[eEk: k 0]; thus E*(E)
is given by Hom E* (E*(E); E*), and admits the dual topological basis e!. It
follows that the composition product dualizes to a noncocommutative co-
product ffiE :E*(E) ! E*(E) E* E*(E) (where E* is taken over the right
action on the left factor), with counit given by projection onto the coeffi-
cients. Together with the left and right units j` and jr, and the antipode OE
induced by interchanging the factors in E*(E), this coproduct turns E*(E)
into a cogroupoid object in the category of E*-algebras. Such an object
generalizes the notion of Hopf algebra, and is known as a Hopf algebroid; for
a detailed discussion, see [22].
We write s for a generic operation in A*E. By virtue of our discussion
above we may interpret s as a spectrum map E ! Sn ^ E, or as an E*-
homomorphism E*(E) ! E*.
When E is replaced by D, we have monomials e` and e!r in D*(D),
induced by tD`and tDrrespectively; the products e`e!rare induced by tD`tDr.
In the cases of interest these products form a basis for D*(D), which is
therefore isomorphic to the polynomial algebra D*[eDj;`; eDk;r: j; k 0].
16 VICTOR M BUHSTABER AND NIGEL RAY
Interpreting the elements of E*(E) as selfmaps of E, we first define the
standard action
(4.1) E*(E) E* E*(X) -! E*(X)
of E*-modules (where E* is taken over the right action on E*(E)) for any
space or spectrum X. We write this action functionally; when X is E it
reduces to the composition product in E*(E). The Cartan formula asserts
that the product map in E*(X) is a homomorphism of left E*(E)-modules
with respect to the action, and was restated by Milnor in the form
X X
(4.2) s(yz) = s0(y)s00(z) where ffiE (s) = s0 s00
for all classes y and z in E*(X). Following Novikov [19], we refer to any
such module with property (4.2)as a Milnor module.
When X is a point (or the sphere spectrum) then (4.1) describes the
action of E*(E) on the coefficient ring E*, and we immediately deduce that
(4.3) s(x) = ~~
for all x in E*.
Given our freeness assumptions we may dualize and conjugate (4.1) in
seven further ways, whose interrelationships are discussed by Boardman
with great erudition. We require four of these (together with a fifth and
sixth which are different), so we select from [2] and [3] without further
comment, relying on the straightforward nature of the algebra. We ignore
all issues concerning signs because our spaces and spectra have no cells in
odd dimensions.
The first of these duals (and our second structure) is the Adams coaction
(4.4) :E*(X) -! E*(E) E* E*(X)
of E*-modules (where E* is taken over the right action on E*(E)) for any
space or spectrum X. This is defined by dualizing the standard action over
E*, and when X is E then reduces to the coproduct ffiE . For each operation
s, the duality may be expressed by
X
(4.5) ~~~~ = ~~~~>;
!
P
where a lies in E*(X) with a! defined by (a) = ! e! a!, and y lies in
E*(X).
If we assume that X is a spectrum (or stable complex), we may interpret
ss*(X ^ E) as X*(E), and consider the isomorphism c: E*(X) ~=X*(E) of
conjugation. Our third structure is the right coaction
(4.6) :X*(E) -! X*(E) E* E*(E)
of right E*-modules (where E* is takenPover the right action of the scalars
on X*(E)); it is evaluated as (ca) = !ca! OE (e!), by conjugating
(4.4). When X is E, then c reduces to OE and becomes ffiE , as before.
Fourthly, the standard action partially dualizes over E* to give the Milnor
coaction
(4.7) ae: E*(X) -! E*(X)b E*E*(E)
COBORDISM, FLAG MANIFOLDS, AND DOUBLES 17
of E*-modules. As Milnor famously observed, the Cartan formulae are en-
capsulated in the fact that ae makes E*(X) a Hopf comodule over E*(E), by
virtue of being an algebra map. For each operation s and each x in E*(X),
the partial duality may be described by
X
(4.8) s(y) = ~~~~y!;
!
P
where y! is defined by ae(y) = ! y! e!; thus y! = e!(y). In view of
the completion required of the tensor product in (4.7), we describe ae more
accurately as a formal coaction. The Chern-Dold character is most naturally
expressed by a simple generalization of (4.7).
A fifth possibility is provided by the left action
E*(E) E* E*(X) -! E*(X)
of E*-modules (where E* is taken over the right action of the scalars on
X*(E)), which is defined by means of spectrum maps in similar fashion to
the standard action (4.1). It is evaluated by partially dualizing the Adams
coaction, for which we write
X
(4.9) s`a = ~~~~a!;
!
with notation as above. The left action satisfies
(4.10) = ~~~~;
where y*: E*(X) ! E*(E) is the homomorphism induced by y, with y and
a as above.
For our sixth and seventh structures we again assume that X is stable,
so the selfmaps of E induce a left action
(4.11) E*(E) E* X*(E) -! X*(E)
(where E* is taken over the right action of the scalars on both factors),
and a right action
X*(E) E* E*(E) -! X*(E)
(where E* is taken over the right action on X*(E)). Neither of these seems
to be discussed explicitly by Boardman, although (4.11)appears regularly
in the literature. It is evaluated by partially dualizing (4.6), for which we
write
X
(4.12) srd = d!~~~~;
!
P
where d lies in X*(E) with (d) = ! d! e!. The two actions are related
according to
(4.13) = <(w)s; d>;
where w lies in X*(E); this should be compared with (4.10), and justifies
the interpretation of (4.11)as a right action (on the left!).
When X is E, then (4.10)may be rewritten as
(4.14) = ~~~~;
18 VICTOR M BUHSTABER AND NIGEL RAY
and (4.13)reduces to the right action of E*(E) on its dual E*(E). We may
evaluate the coproduct ffiE (d) as
X X
(4.15) e!;rd e! = OE (e!) e!;`d;
! !
given any d in E*(E).
It is important to record how the elements ek and the orientation class
xE are intertwined by certain of these actions and coactions.
Proposition 4.16. We have that
X
ae(xE ) = (xE )k+1 ek
k0
in E*(CP 1 )b E*E*(E).
Proof. The induced homomorphism xE*:E*+2(CP 1 ) ! E*(E) acts such
that xE*(fiEk+1) = ek, by definition of the elements e!. Dualizing, we obtain
X
(4.17) s(xE ) = ~~~~(xE )k+1;
k0
__
and the formula follows from (4.8). |__|
By a simple extension of (4.17), the entire algebra AE* may be faithfully
represented by its action on the bottom cell of the smash product of infinitely
many copies of CP 1 .
Corollary 4.18. The coproduct and antipode of the Hopf algebroid E*(E)
are given by
X
ffiE (en) = (e)k+1n-k ek and OE (en) = (e)-(n+1)n
k0
respectively.
Proof. Since ae is a coaction, we have that ae 1(ae(xE )) = 1 ffiE (ae(xE )) *
*as
maps E*(CP 1 ) ! E*(CP 1 )b E*E*(E) E* E*(E), and the formula for ffiE
follows. The properties of the antipode, coupled with Lagrange inversion,_
yield OE immediately. |__|
We remark in passing that Boardman uses the formulae for ae(xE ) to define
the elements ek.
When E is replaced by D, we find that
X
ae(xD`) = (xD`)k+1 ek;` 1 and
k0
(4.19) X
ae(xDr) = (xDr)k+1 1 ek;r
k0
in D*(CP 1 )b D*D*(D). These two expressions are linked by the series
xDr = g(xD`) of (3.10), and were exploited by the first author in [8]. In
similar vein, ffiD and OD on eDn;`and eDn;rare given by the left and right
forms of Corollary 4.18.
Armed with these results, we now consider the universal example MU .
COBORDISM, FLAG MANIFOLDS, AND DOUBLES 19
As described in x3, the operations e! are written s! in A*MU, and are
known as the Landweber-Novikov operations; dually, the elements e! reduce
to b! in AMU*. We consider the integral spans S* and S* of the s! and b!
respectively, so that A*MU ~=U* S* and AMU* ~=U* S* as U* -modules,
where S* is the polynomial algebra Z[bk : k 0]. The U* -duality between
A*MU and AMU* therefore restricts to an integral duality between S* and S*,
for which no topological considerations are necessary because S* has finite
type.
The formulae of Proposition 4.16 show that S* is closed with respect to
the coproduct and antipode of AMU*, whilst the left unit and the counit
survive with respect to Z. Therefore S* is a Hopf subalgebra of the Hopf
algebroid. Duality then ensures that S* is also a Hopf algebra, with respect
to composition of operations and the Cartan formula
X
(4.20) ffi(s!) = s!0 s!00;
!0+!00=!
which is dual to the product of monomials. Of course S* is the Landweber-
Novikov algebra. Alternatively, and following the original constructions, we
may use the action of S* on *U (^1 CP 1 ) to prove directly that S* is a Hopf
algebra. Many of our actions and coactions restrict to S* and S* and will be
important below. We emphasize that A*MU has no U* -linear antipode, and
that the antipode in S* is induced from the antipode in S* by Z-duality.
Choosing E and X to be MU in (4.13)and (4.14)provides the left and
right action of A*MU on its dual. Explicitly, if s and y lie in A*MU and b in
AMU*, then
(4.21) = ~~~~ and = :
Alternatively, by appealing to (4.9)and (4.12)we may write
X X
(4.22) s`b = ~~~~b00 and srb = ~~~~b0;
P
where ffi(b) = b0 b00. By restriction we obtain identical formulae for the
left and right actions of S* on AMU* and on S*. In the latter case, Z-duality
allows us to rewrite the left action as
= ;
thereby (at last) according it equivalent status to the right action.
The adjoint actions of S* on AMU* and S* are similarly defined by
X
= and
(4.23) X
ad (s)(b) = b00;
which give rise to the adjoint Milnor module structure on AMU* and S*.
By way of example, we recallPfrom Corollary 4.18 that the diagonal for
S* is given by ffi(bn) = k0 (b)k+1n-k bk; (4.15)therefore yields
sffl(k);`bn(=k - n - 1)bn-k ; sffl(k);rbn = (b)k+1n-k
(4.24)
and ad (sffl(k))bn = (k - n - 1)bn-k + (b)k+1n-k
for all 0 k n.
20 VICTOR M BUHSTABER AND NIGEL RAY
The Thom isomorphisms (3.4) ensure that the action of S* on the co-
efficients U* has certain special geometrical attributes. By restricting the
arguments of Proposition 3.5, a representative for the action of the right
unit
(4.25) jr: U* - ! AMU* ~=U* (BU +)
on the cobordism class of a U-manifold Mk is given by the singular U-
manifold :Mk ! BU. Since the operation s! corresponds to the Chern
class c! under A*MU ~=*U (BU +), we deduce from (4.3)that
s!(Mk) = ;
where oe in Uk(Mk) is the canonical orientation class represented by the
identity map on Mk. In other words, s!(Mk) in Uk-2|!|is represented by
the domain of the Poincare dual of the the normal Chern class c!().
Following Novikov we use this action to illuminate the product structure
in A*MU, within which we have already identified the subalgebras U* and S*.
It therefore suffices to describe the commutation rule for expressing products
of the form sx, where s and x lie in S* and U* respectively. Recalling (4.1)
and (4.2)we obtain
X
(4.26) sx = s0(x)s00;
and write A*MU = U* S* for the resulting algebra. We may construct an
algebra in this fashion from any Milnor module structure, and we refer
to it as the associated operator double. It is an important special case of
Sweedler's smash product [27]. The coproduct in A*MU is obtained from
(4.20) by U* -linearity.
We may describe the coproduct for AMU* in terms of the polynomial alge-
bra U* S*, simply by using U* -linearity to extend the coproduct for S*.
Dually, the form of (4.26)is governed by the Adams coaction, as expressed
in (4.5).
There are other ways of interpreting the algebra S*. For example, the
projection tH*: U* (MU ) ! H*(MU ) restricts to an isomorphism on S*,
with tH*(b!) = (bH )!. This isomorphism often appears implicitly in the
literature, although the induced coproduct and antipode in H*(MU ) are
purely algebraic. A second interpretation, for which we recall the canonical
isomorphism DU* ~=AMU* of (2.6), is crucial.
Proposition 4.27. The subalgebra G* of DU* is identified with the dual of
the Landweber-Novikov algebra S* in AMU* under the canonical isomorphism.
Proof. By appealing to Proposition 3.8, it suffices to show that the Thom
isomorphism AMU* ~=U* (BU +) satisfies bk 7! fik. This follows by definition,
and ensures (by multiplicativity) that monomials g! and b! are identified_
for all !. |__|
We now consider operations in double complex cobordism theory, and
their interaction with the Landweber-Novikov algebra. For this purpose we
apply our general theory in the case when E is DU, from which we imme-
diately obtain the algebra of DU-operations A*DU, and of DU-cooperations
ADU*. These act and coact according to the eightfold way.
COBORDISM, FLAG MANIFOLDS, AND DOUBLES 21
Identifying D with MU ^ MU , we note that an element s of A*MU yields
operations s 1 and 1 s in A*DU by action on the appropriate factor; this
description is consistent with our choice of DU* -basis elements s s! from
amongst the four possibilities in (3.13),. We refer to the operations s 1
and 1s! as left and right Landweber-Novikov operations, and observe that
they commute by construction. Thus A*DU contains the subalgebra S* S*,
and ADU* contains the subalgebra S* S* ~= Z[bj 1; 1 bk : j; k 0].
Occasionally we write S* 1 and S* 1 as S*`and S*;`respectively, with
similar conventions on the right. As before, the DU* -duality between A*DU
and ADU* restricts to an integral duality between S* S* and S* S*. Since
S*S* is a Hopf subalgebra of the Hopf algebroid ADU*, this duality ensures
that S*S* is also a Hopf algebra with respect to composition of operations.
The coproduct is dual to the product of monomials, being given by left and
right Cartan formulae of the form (4.20), and the antipode is induced from
S* S* by Z-duality. These structures are, of course, identical with those
obtained by forming the square of each of the Hopf algebras S* and S*.
The coproduct ffi :S* ! S* S* provides a third (and extremely impor-
tant) diagonal embedding of S* in A*DU.
Writing E as DU and X as a point (or the sphere spectrum) in (4.1)
yields the action of A*DU on the coefficient ring DU* ; the action of S* S*
follows by restriction. Both of these give rise to a Milnor module structure,
so that we may express A*DU as the operator double DU* (S* S*). The
actions are closely related to those of (4.21)under the canonical isomorphism
DU* ~=AMU*.
Proposition 4.28. The canonical isomorphism identifies the actions of S*`
and S*ron DU* with the left and right actions of S* on AMU* respectively.
Proof. This follows from the definitions by identifying the respective actions
in terms of maps of spectra; thus, on an element x in ssk(MU ^ MU ), the
action of a left operation s` and the left action of s are both represented by
Sk -x!MU ^ MU -s^1-!MU ^ MU :
__
For the right actions, we replace s ^ 1 with 1 ^ s. |__|
Corollary 4.29. The subalgebra G* of DU* is closed under the action of
the subalgebra S* S* of A*DU.
__
Proof. We apply Proposition 4.27, and the result is immediate. |__|
We shall say more about Corollary 4.29 in x6, where we give an interpretation
in terms of the geometry of flag manifolds.
We may combine Proposition 4.28 and Corollary 4.29 to ensure that the
diagonal action of S* on DU* and G* is identified with the adjoint action
on S*. This result was established in [8] by appeal to the coaction ae as in
(4.19), and we now restate its consequences; the proofs above are valid for
any spectrum of the form E ^ F .
Since S* is cocommutative, we utilize Novikov's construction [19] (as also
described in [17]) of D(H*) as the operator double S*S* with respect to
the adjoint action (4.23)of S* on its dual. Our realization of the quantum
double D(S*) follows.
22 VICTOR M BUHSTABER AND NIGEL RAY
Theorem 4.30 ([8]). The algebra of operations A*DU contains a subalgebra
isomorphic to D(S*).
Proof. As explained above, we may express A*DU as the operator double
DU* (S* S*) with respect to the standard action of S* S* on DU* . The
action of the diagonal subalgebra S* restricts to G*, and is identified with
the adjoint action on S* by Corollary 4.29. Since the operator double S*G* __
is a subalgebra of A*DU, the result follows. |__|
It follows directly from the definitions, coupled with the formulae (4.24),
that the commutation law in D(S*) obeys
(4.31) sffl(k)bn = (k - n - 1)bn-k + (b)k+1n-k+ bnsffl(k)
for all 0 k n.
By analogy with (4.25)we describe the action of S* S* on DU* geo-
metrically. A representative for the action of the right unit
jr: DU* -! ADU* ~=DU* (BU x BU + )
on the cobordism class of a double U-manifold Mk is given by the singular
double U-manifold ` x r: Mk ! BU x BU . Since the operation s s!
corresponds to the Chern class c c! under A*DU ~=*DU (BU x BU +), we
conclude from (4.3)that
(4.32) s s!(Mk) = ;
where oe in DUk(Mk) is the canonical orientation class represented by the
identity map on Mk.
We deduce from (4.32)that the left, right, and diagonal actions of s! on
Mk give , , and
X
!0+!00=!
respectively, in DUk-2|!. In conjunction with Proposition 4.28, these formulae
yield a geometrical realization of the three actions of S* on AMU*; we take
the double U-cobordismPclass of Mk, form the Poincare duals of c!;`(`),
c!;r(r), and !0+!00=!c!0;`(`)c!00;r(r) respectively, and record the double
U-cobordism class of the domain. We shall implement this procedure in
terms of bounded flag manifolds in x6.
5. Bounded flag manifolds
In this section we introduce our family of bounded flag manifolds, and
discuss their topology in terms of a cellular calculus which is intimately
related to the Schubert calculus for classic flag manifolds. Our description
is couched in terms of nonsingular subvarieties, anticipating applications to
cobordism in the next section. We also invest the bounded flag manifolds
with certain canonical U- and double U-structures, and so relate them to
our earlier constructions in DU* . Much of our notation differs considerably
from that introduced by the second author in [24].
We shall follow combinatorial convention by writing [n] for the set of
natural numbers {1; 2; : :;:n}, equipped with the standard linear ordering
< . Every interval in the poset [n] has the form [a; b] for some 1 a < b n,
COBORDISM, FLAG MANIFOLDS, AND DOUBLES 23
and consists of all m satisfying a m b; our convention therefore dictates
that we abbreviate [1; b] to [b]. It is occasionally convenient to interpret
[0] as the empty set, and [1] as the natural numbers. We work in the
context of the Boolean algebra B(n) of finite subsets of [n], ordered by
inclusion. We decompose each such subset Q [n] into maximal subintervals
I(1) [ . .[.I(s), where I(j) = [a(j); b(j)] for 1 j s, and assign to Q the
monomial b!, where !i records the number of intervals I(j) of cardinality
i for each 1 i n; we refer to ! as the type of Q, noting that it is
independent of the choice of n. We display the elements of Q in increasing
order as {qi : 1 i d}, and abbreviate the complement [n] \ Q to Q0. We
also write I(j)+ for the subinterval [a(j); b(j) + 1] of [n + 1], and Q^ for
Q [ {n + 1}. It is occasionally convenient to set b(0) to 0 and a(s + 1) to
n + 1.
We begin by recalling standard constructions of complex flag manifolds
and some of their simple properties, for which a helpful reference is [12]. We
work in an ambient complex inner product space Zn+1 , which we assume to
be invested with a preferred orthonormal basis z1, : :,:zn+1 , and we write
ZE for the subspace spanned by the vectors {ze : e 2 E}, where E [n + 1].
We abbreviate Z[a;b]to Za;b(and Z[b]to Zb) for each 1 a < b n + 1,
and write CP (ZE ) for the projective space of lines in ZE . We let V - U
denote the orthogonal complement of U in V for any subspaces U < V of
Zn+1 , and we regularly abuse notation by writing 0 for the subspace which
consists only of the zero vector. A complete flag V in Zn+1 is a sequence of
proper subspaces
0 = V0 < V1 < . .<.Vi < . .<.Vn < Vn+1 = Zn+1 ;
of which the standard flag Z0 < . .<.Zi < . .<.Zn+1 is a specific example.
The flag manifold F (Zn+1 ) is the set of all flags in Zn+1 , topologized ap-
propriately. Since the unitary group U(n + 1) acts transitively on F (Zn+1 )
in such a way that the stabilizer of the standard flag is the maximal torus
T , we may identify F (Zn+1 ) with the coset space U(n + 1)=T ; in this guise,
F (Zn+1 ) acquires the quotient topology.
In fact the flag manifold has many extra properties. It is a nonsingular
complex projective algebraic variety of dimension n+12, and is therefore a
closed complex manifold. It has a CW-structure whose cells are even dimen-
sional and may be described in terms of the Bruhat decomposition, which
indexes them by elements ff of the symmetric group n+1 (the Weyl group
of U(n+1)), and partially orders them by the decomposition of ff into prod-
ucts of transpositions. The closure of each Bruhat cell effis an algebraic
subvariety, generally singular, known as the Schubert variety Xff. Whether
considered as cells or subvarieties, these subspaces define a basis for the int*
*e-
gral homology H*(F (Zn+1 )), and therefore also for the integral cohomology
H*(F (Zn+1 )) which is simply the integral dual. The manipulation of cup
and cap products and Poincare duality with respect to these bases is known
as the Schubert calculus for F (Zn+1 ), and has long been a source of delight
to geometers and combinatorialists.
An alternative description of H*(F (Zn+1 )) is provided by Borel's com-
putations with the Serre spectral sequence. The canonical torus bundle
over U(n + 1)=T is classified by a map U(n + 1)=T ! BT , which induces
24 VICTOR M BUHSTABER AND NIGEL RAY
the characteristic homomorphism H*(BT ) ! H*(U(n + 1)=T ) in integral
cohomology. Noting that H*(BT ) is a polynomial algebra over Z on 2-
dimensional classes xi, where 1 i n + 1, Borel identifies H*(F (Zn+1 ))
with the ring of coinvariants under the action of the Weyl group, defined as
the quotient Z[x1; : :;:xn+1 ]=J, where J is the ideal generated by all sym-
metric polynomials. With respect to this identification, xi is the first Chern
class of the line bundle over F (Zn+1 ) obtained by associating Vi- Vi-1 to
each flag V .
The interaction between the Schubert and Borel descriptions of the coho-
mology of F (Zn+1 ) is a fascinating area of combinatorial algebra and has led
to a burgeoning literature on the subject of Schubert polynomials, beautifully
surveyed in MacDonald's book [14]. The entire study may be generalized to
quotients such as G=B, where G is a semisimple algebraic group over a field
k, and B is an arbitrary Borel subgroup.
We call a flag U in Zn+1 bounded if each i-dimensional component Ui
contains the first i-1 basis vectors z1, : :,:zi-1, or equivalently, if Zi-1 < *
*Ui
for every 1 i n + 1. We define the bounded flag manifold B(Zn+1 ) to be
the set of all bounded flags in Zn+1 , topologized as a subspace of F (Zn+1 );
its complex manifold structure arises by choosing a neighbourhood of U
to consist of all bounded flags T satisfying Ti \ U?i = 0. It is simple to
check that, as i decreases, this condition restricts each proper subspace Ti
to a single degree of freedom and defines a chart of dimension n. Clearly
B(Z2) is isomorphic to the projective line CP (Z2) with the standard complex
structure, whilst B(Z1) consists solely of the trivial flag. We occasionally
abbreviate B(Zn+1 ) to Bn, in recognition of its dimension.
We shall devote the remainder of this section to the topology of bounded
flag manifolds and a discussion of their U- and double U-structures.
There is a map ph :B(Zn+1 ) ! B(Zh+1;n+1) for each 1 h n, defined
by factoring out Zh. Thus ph(U) is given by
0 < Uh+1 - Zh < . .<.Ui- Zh < . .<.Un - Zh < Zh+1;n+1
for each bounded flag U in Zn+1 . Since Zi-1 < Ui for all 1 i n + 1,
we deduce that Zh+1;i-1< Ui- Zh for all i > h + 1, ensuring that ph(U)
is indeed bounded. We may readily check that ph is the projection of a
fiber bundle, with fiber B(Zh+1). In particular, p1 has fiber the projective
line CP (Z2), and so after n - 1 applications we may exhibit B(Zn+1 ) as an
iterated bundle
(5.1) B(Zn+1 ) ! . .!.B(Zh;n+1) ! . .!.B(Zn;n+1)
over B(Zn;n+1), where the fiber of each map is isomorphic to CP 1. This
construction was used by the second author in [24].
We define maps qh and rh :B(Zn+1 ) ! CP (Zh;n+1) by letting qh(U)
and rh(U) be the respective lines Uh - Zh-1 and Uh+1 - Uh, for each
1 h n. We remark that qh = q1 . ph-1 and rh = r1 . ph-1 for all h,
and that the appropriate qh and rh may be assembled into maps qQ and
rQ :B(Zn+1 ) ! xQ CP (Zh;n+1), where h varies over an arbitrary subset
Q of [n]. In particular, q[n]and r[n]are embeddings, which associate to
each flag U the n-tuple (U1; : :;:Uh - Zh-1; : :;:Un - Zn-1 ) and the n-tuple
COBORDISM, FLAG MANIFOLDS, AND DOUBLES 25
(U2-U1; : :;:Uh+1 -Uh; : :;:Zn+1 -Un) respectively, and lead to descriptions
of B(Zn+1 ) as a projective algebraic variety.
We proceed by analogy with the Schubert calculus for F (Zn+1 ). To every
flag U in B(Zn+1 ) we assign the support S(U), given by {j 2 [n] : Uj 6= Zj},
and consider the subspace
eQ = {U 2 B(Zn+1 ) : S(U) = Q}
for each Q in the Boolean algebra B(n). For example, e; is the singleton
consisting of the standard flag.
Lemma 5.2. For all nonempty Q, the subspace eQ B(Zn+1 ) is an open
cell of dimension 2|Q|, whose closure XQ is the union of all eR for which
R Q in B(n).
Proof. If Q = [jI(j), then eQ is homeomorphic to the cartesian product
xjeI(j), so it suffices to assume that Q is an interval [a; b]. If U lies in e[*
*a;b]
then Ua-1 = Za-1 and Ub+1 = Zb+1 certainly both hold; thus e[a;b]consists
of those flags U for which qj(U) is a fixed line L in CP (Za;b+1) \ CP (Za;b)
for all a j b. Therefore e[a;b]is a 2(b - a + 1)-cell, as sought. Obviously
eR X[a;b]for each R Q, so it remains only to observe that the limit of a
sequence of flags in e[a;b]cannot have fewer components satisfying Uj = Zj,_
and must therefore lie in eR for some R [a; b]. |__|
Clearly X[n]is B(Zn+1 ), so that Lemma 5.2 provides a CW decomposition
for Bn with 2n cells.
We now prove that all the subvarieties XQ are nonsingular, in contrast to
the situation for F (Zn+1 ).
Proposition 5.3. For any Q [n], the subvariety XQ is diffeomorphic to
the cartesian product xjB(ZI(j)+).
Proof. We may define a smooth embedding iQ : xj B(ZI(j)+) ! B(Zn+1 )
by choosing the components of iQ (U(1); : :;:U(s)) to be
(
Za(j)-1 U(j)i if k = a(j) + i - 1 in I(j)
(5.4) Tk =
Zk if k 2 [n + 1] \;Q
where U(j)i < ZI(j)+for each 1 i b(j) - a(j) + 1; the resulting flag
is indeed bounded, since Za(j);a(j)+i-1< U(j)i holds for all such i and
1 j s. Any flag T in B(Zn+1 ) for which S(T ) Q must be of the form __
(5.4), so that iQ has image XQ , as required. |__|
We may therefore interpret the set
X (n) = {XQ : Q 2 B(n)}
as a Boolean algebra of nonsingular subvarieties of B(Zn+1 ), ordered by
inclusion, on which the support function S :X (n) ! B(n) induces an iso-
morphism of Boolean algebras. Moreover, whenever Q has type ! then XQ
is isomorphic to the cartesian product B!11B!22: :B:!nn, and so may be ab-
breviated to B! . In this important sense, S preserves types. We note that
the complex dimension |Q| of XQ may be written as |!|.
The following quartet of lemmas forms the core of our calculus, and is
central to computations in x6.
26 VICTOR M BUHSTABER AND NIGEL RAY
Lemma 5.5. The map rQ0: B(Zn+1 ) ! xQ0CP (Zh;n+1) is transverse to
the subvariety xQ0CP (Zh+1;n+1), whose inverse image is XQ .
Proof. Let T be a flag in B(Zn+1 ). Then rh(T ) lies CP (Zh+1;n+1) if and
only if Th+1 = Th Lh for some line Lh in Zh+1;n+1. Since Zh < Th+1, this
condition is equivalent to requiring that Th = Zh, and the proof is completed_
by allowing h to range over Q0. |__|
Lemma 5.6. The map qQ0: B(Zn+1 ) ! xQ0CP (Zh;n+1) is transverse to
the subvariety xQ0CP (Zh+1;n+1), whose inverse image is diffeomorphic to
B(ZQ^ ).
Proof. Let T be a flag in B(Zn+1 ) such that qh(T ) lies CP (Zh+1;n+1), which
occurs if and only if Th = Zh-1 Lh for some line Lh in Zh+1;n+1. Whenever
this equation holds for all h in some interval [a; b], we deduce that Lh actual*
*ly
lies in Zb+1;n+1. Thus we may describe T globally by
Tk = Z[k-1]\Q Ui;
where Ui lies in ZQ^ , and i is k - |[k - 1] \ Q|. Clearly Ui-1 < Ui and
Z{q1;:::;qi-1}< Ui for all appropriate i, so that U lies in B(ZQ^ ). We may
now identify the required inverse image with the image of the natural smooth_
embedding jQ :B(ZQ^ ) ! B(Zn+1 ), as sought. |__|
We therefore define YQ to consist of all flags T for which the line Th-Zh-1
lies in ZQ^ for every h in Q0. Since Y[n]is B(Zn+1 ) (and Y; is the singleton
standard flag), the set
Y(n) = {YQ : Q 2 B(n)}
is also a Boolean algebra of nonsingular subvarieties. In this instance, how-
ever, YQ is isomorphic to Bk whenever Q has cardinality k, irrespective of
type. We may consider XQ and YQ0 to be complementary, insofar as the
supports of the constituent flags satisfy S(T ) Q and Q S(T ) respec-
tively.
Lemma 5.7. For any 1 m n - h, the map qh :B(Zn+1 ) ! CP (Zh;n+1)
is transverse to the subvariety CP (Zh+m;n+1 ), whose inverse image is dif-
feomorphic to Y[h;h+m-1]0.
Proof. Let T be a flag in B(Zn+1 ) such that qh(T ) lies CP (Zh+m;n+1 ), which
occurs if and only if Th = Zh-1 Lh for some line Lh in Zh+m;n+1 . Following
the proof of Lemma 5.6 we immediately identify the required inverse image__
with Y[h-1][[h+m;n], as sought. |__|
Lemma 5.8. The following intersections in B(Zn+1 ) are transverse:
XQ \ XR = XQ\R and YQ \ YR = YQ\R whenever Q [ R = [n];
(
XQ\R; R if Q [ R = [n]
and XQ \ YR =
; otherwise;
where XQ\R; R denotes the submanifold XQ\R B(ZR^ ). Moreover, m
copies of Y{h}0may be made self-transverse so that
Y{h}0\ . .\.Y{h}0= Y[h;h+m-1]0
for each 1 h n and 1 m n - h.
COBORDISM, FLAG MANIFOLDS, AND DOUBLES 27
Proof. The first three formulae follow directly from the definitions, and di-
mensional considerations ensure that the intersections are transverse. The
manifold XQ\R; R is diffeomorphic to x jYR0(j)as a submanifold of B(Zn+1 ),
where Q = [jI(j) and R0(j) = R0\ I(j) for each 1 j s.
Since Y{h}0is defined by the single constraint Uh = Zh-1 Lh, where Lh
is a line in Zh+1;n+1, we may deform the embedding j{h}0(through smooth
embeddings, in fact) to m - 1 further embeddings in which the Lh is con-
strained to lie in Z[h;n+1]\{h+i-1}, for each 2 i m. The intersection of the
m resulting images is determined by the single constraint Lh < Zh+m;n+1 ,_
and the result follows by applying Lemma 5.7. |__|
We conclude this section with a study of the U- and double U-structures
on B(Zn+1 ), for which a few bundle-theoretic preliminaries are required.
For each 1 i n we consider the complex line bundles fli and aei,
classified respectively by the maps qi and ri. It is consistent to take fl0,
fln+1 , and ae0 to be 0, C and fl1 respectively, from which we deduce that
(5.9) fli aei aei+1 . . .aen ~=C n-i+2
for every i. Since we may use (5.1)as in [24] to obtain an expression of the
form o R ~=(n+1i=2fli)R for the tangent bundle of B(Zn+1 ), so (5.9)leads
to an isomorphism ~=ni=2(i-1)aei. We refer to the resulting U-structure as
the basic U-structure on B(Zn+1 ). We emphasise that these isomorphisms
are of real bundles only, and therefore that the basic U-structure does not
arise from the underlying complex algebraic variety. On B(Z2), for example,
the basic U-structure is that of a 2-sphere S2, rather than CP 1. Indeed, the
basic U-structure on B(Zn+1 ) extends over the 3-disc bundle associated to
fl1 R for all values of n, soLthat B(Zn+1 ) represents zero in U2n.
If we split so that ` is ni=1iaei and r is fl1 (appealing to (5.9)), we
again refer to the resulting double U-structure as basic; equivalently, we
may rewrite ` stably as -(fl1 . . .fln). The basic double U-structure
does not bound, however, as we shall see in Proposition 5.10. Given any
cartesian product of manifolds B(Zn+1 ), we also refer to the product of basic
structures as basic.
We may now formulate the fundamental connection between bounded flag
manifolds and the Landweber-Novikov algebra.
Proposition 5.10. With the basic double U-structure, B(Zn+1 ) represents
gn in DU*; if the left and right components of are interchanged, it repre-
sents __gn.
Proof. Applying Proposition 3.5, the image of the double U-cobordism class
of B(Zn+1 ) is represented by the singular U-manifold fl1: B(Zn+1 ) ! BU
in U2n(BU +) under the isomorphism h. Since fl1 lifts to CP 1 this class is
fin, as proven in [24]. Appealing to Proposition 3.8 completes the proof for__
gn, and the result follows for __gnby applying the involution O. |__|
Corollary 5.11. Under the canonical isomorphism G* ~=S*, the cobordism
classes of the basic double U-manifolds XQ give an additive basis for the
dual of the Landweber-Novikov algebra, as Q ranges over finite subsets of
[1].
28 VICTOR M BUHSTABER AND NIGEL RAY
Proof. By Proposition 4.27, the canonical isomorphism identifies the mono-
mials g! and b!. Applying Proposition 5.10 to cartesian products, we deduce
that XQ (and therefore B! ) represents g! whenever XQ carries the basic_
double U-structure and Q has type !. The result follows. |__|
Henceforth we shall insist that Bn denotes B(Zn+1 ) (or any isomorph) only
when equipped with the basic double U-structure.
Proposition 5.12. Both X (n) and Y(n) are Boolean algebras of basic U-
submanifolds, in which the intersection formulae of Lemma 5.8 respect the
basic U-structures.
Proof. It suffices to prove that the pullbacks in Lemmas 5.5, 5.6 and 5.7 are
compatible with the basic U-structures. Beginning with Lemma 5.5, we note
that whenever aeh over B(Zn+1 ) is restricted by iQ to a factor B(ZI(j)+), we
obtain aek+1 if h = a(j) + k lies in I(j) and fl1 if h = a(j) - 1 (unless 1 2 Q*
*);
for all other values of h, the restriction is trivial. Since the construction
of Lemma 5.5 identifies (iQ ) with the restriction of haeh as h ranges over
Q0, we infer an isomorphism (iQ ) ~= (xjfl1) C n-j-|Q|over XQ (unless
1 2 Q, in which case the first fl1 is trivial). Appealing to (5.9), we then
verify that this is compatible with the basic structures in the isomorphism
XQ ~=(iQ )*B(Zn+1) (iQ ), as claimed. The proofs for Lemmas 5.6 and
5.7 are similar, noting that the restriction of aeh to YQ is aek+1 if h = a(j) *
*+ k
lies in Q, and is trivial otherwise, and that the restriction of flh is flk if *
*h = qk
lies in Q, and is flk+1 if qk is the greatest element of Q for which h > qk
(meaning fl1 if h < q1, and the trivial bundle if h > qk for all k). Since the
construction of Lemma 5.6 identifies (jQ ) with the restriction of hflh as
h ranges over Q0, we infer an isomorphism
s+1Mi j
(5.13) (jQ ) ~= a(j) - b(j - 1) - 1 flc(j)
j=1
P j-1
over YQ , where c(j) = j + i=0(b(i) - a(i)). This isomorphism is also
compatible with the basic structures in YQ ~= (jQ )*B(Zn+1) (jQ ), once__
more by appeal to (5.9). |__|
The corresponding results for double U-structures are more subtle, since
we are free to choose our splitting of (iQ ) and (jQ ) into left and right
components.
Corollary 5.14. The same results hold for double U-structures with respect
to the splittings (iQ )` = 0 and (iQ )r = (iQ ), and (jQ )` = (jQ ) and
(jQ )r = 0.
Proof. One extra fact is required in the calculation for iQ , namely that fl1__
on B(Zn+1 ) restricts trivially to XQ (or to fl1 if 1 2 Q). |__|
At this juncture we may identify the inclusions of XQ in F (Zn+1 ) with cer-
tain of the desingularizations introduced by Bott and Samelson [4]; for exam-
ple, X[n]provides the desingularization of the Schubert variety X(n+1;1;2;:::;n*
*).
Moreover, the corresponding U-cobordism classes form the cornerstone of
Bressler and Evens's calculus for *U (F (Zn+1 )). In both of these applica-
tions, however, the underlying complex manifold structures suffice. The
COBORDISM, FLAG MANIFOLDS, AND DOUBLES 29
basic U-structures become vital when investigating the Landweber-Novikov
algebra (and could also have been used in [5], although a different calculus
would result). We leave the details to interested readers.
6. Computations and formulae
In this section we study the normal characteristic numbers of bounded
flag manifolds, and deduce formulae for the actions of various cohomology
operations on the corresponding bordism classes. These allow us to provide
our promised geometrical realization of many of the algebraic structures of
the Landweber-Novikov algebra, and its dual and quantum double.
We begin by recalling the CW decomposition of B(Zn+1 ) resulting from
Lemma 5.2, and noting that the cells eQ define a basis for the cellular
chain complex. Since they occur only in even dimensions, the correspond-
ing homology classes xHQ form a Z-basis for the integral homology groups
H*(B(Zn+1 )) as Q ranges over B(n). Applying Hom Z, we obtain a dual
basis Hd (xHQ) for the integral cohomology groups H*(B(Zn+1 )); we delay
clarifying the cup product structure until after Theorem 6.2 below.
We introduce the complex bordism classes xQ and yQ in U2|Q|(B(Zn+1 )),
represented respectively by the inclusions iQ and jQ of the subvarieties XQ
and YQ with their basic U-structures. By construction, the fundamental
class in H2|Q|(XQ ) maps to xHQ in H2|Q|(B(Zn+1 )) under iQ ; thus xQ maps
to xHQunder the homomorphism U* (B(Zn+1 )) ! H*(B(Zn+1 )) induced (as
described in x3) by the Thom class tH . The Atiyah-Hirzebruch spectral
sequence for U* (B(Zn+1 )) therefore collapses, and the classes xQ form an
U* -basis as Q ranges over B(n). The classes x[n]and y[n]coincide, since
they are both represented by the identity map. They constitute the basic
fundamental class in U2n(B(Zn+1 )), with respect to which the Poincare
duality isomorphism is given by
Pd (w) = w \ x[n]
in U2(n-d)(B(Zn+1 )), for any w in 2dU(B(Zn+1 )).
An alternative source of elements in 2U(B(Zn+1 )) is provided by the
Chern classes
xi = c1(fli) and yi = c1(aei)
for each 1 i n. It follows from (5.9)that
(6.1) xi = -yi- yi+1 - . .-.yn
Q Q
for every i. Given Q [n], we write Q xh as xQ and Q yh as yQ in
2|Q|U(B(Zn+1 )), where h ranges over Q in both products.
We may now discuss the implications of our intersection results of Lemma
5.8 for the structure of *U(B(Zn+1 )). It is convenient (but by no means
necessary) to use Quillen's geometrical interpretation of cobordism classes,
which provides a particularly succinct description of cup and cap products
and Poincare duality, and is conveniently summarized in [5].
Theorem 6.2. The complex bordism and cobordism of B(Zn+1 ) satisfy
(1)Pd (xQ0) = yQ and Pd (yQ0) = xQ ;
30 VICTOR M BUHSTABER AND NIGEL RAY
(2)the elements {yQ : Q [n]} form an U* -basis for U* (B(Zn+1 ));
(3)Hd (xQ ) = xQ and Hd (yQ ) = yQ ;
(4)there is an isomorphism of rings
*U(B(Zn+1 )) ~=U* [x1; : :;:xn]=(x2i= xixi+1);
where i ranges over [n] and xn+1 is interpreted as 0.
Proof. For (1), we apply Lemma 5.6 and Proposition 5.12 to deduce that
0|
xQ0 in 2|QU(B(Zn+1 )) is the pullback of the Thom class under the collapse
map onto M((jQ )). Hence xQ0 is represented geometrically by the inclu-
sion jQ :YQ ! B(Zn+1 ), and therefore Pd (xQ0) is represented by the same
singular U-manifold in 2|Q|(B(Zn+1 )). Thus Pd (xQ0) = yQ . An identical
method works for Pd (yQ0), by applying Lemma 5.5. For (2), we have already
shown that the xQ form an U* -basis for U* (B(Zn+1 )). Thus by (1) the yQ
form a basis for *U (B(Zn+1 )), and therefore so do the xQ by (6.1); the proof
is concluded by appealing to (1) once more. To establish (3), we remark that
the cap product xQ \ xR is represented geometrically by the fiber product of
jQ0 and iR , and is therefore computed by the intersection theory of Lemma
5.8. Bearing in mind the crucial fact that each basic U-structure bounds
(except in dimension zero!), we obtain
(6.3) = ffiQ;R
and therefore that Hd (xQ ) = xQ , as sought. The result for Hd (yQ ) follows
similarly. To prove (4) we note that it suffices to obtain the product formula
x2i= xixi+1, since we have already demonstrated that the monomials xQ
form a basis in (2). Now xi and xi+1 are represented geometrically by
Y{i}0and Y{i+1}0respectively, and products are represented by intersections;
according to Lemma 5.8 (with m = 2), both x2iand xixi+1 are therefore
represented by the same subvariety Y{i;i+1}0, so long as 1 i < n. When
__
i = n we note that xn pulls back from CP 1, so that x2n= 0, as required. |__|
For any Q [n], we obtain the corresponding structures for the com-
plex bordism and cobordism of XQ by applying the K"unneth formulae to
Theorem 6.2. Using the same notation as in B(Zn+1 ) for any cohomology
class which restricts along (or homology class which factors through) the
inclusion iQ , we deduce, for example, a ring isomorphism
(6.4) *U(XQ ) ~=U* [xi : i 2 Q]=(x2i= xixi+1);
where xi is interpreted as 0 for all i =2Q.
The relationship between the classes xi and yi in *U(B(Zn+1 )) is de-
scribed by (6.1), but may be established directly by appeal to the third
formula of Lemma 5.8, as in the proof of Theorem 6.2; for example, we
deduce immediately that xiyi = 0 for all 1 i n. When applied with
arbitrary m, the fourth formula of Lemma 5.8 simply iterates the quadratic
relations, and produces nothing new.
Of course we may extend the results of Theorem 6.2 and its corollaries to
any complex oriented spectrum E. We define xEQand yEQin E2|Q|(B(Zn+1 ))
to be the respective images of the E-homology fundamental classes of XQ
and YQ , and xQEand yQEin E2|Q|(B(Zn+1 )) to be the appropriate monomials
COBORDISM, FLAG MANIFOLDS, AND DOUBLES 31
in the E-cohomology Chern classes of0the fli and aei0respectively. We apply
the Thom class tE to deduce that xQE and yEQ, and yQE and xEQ, are Poincare
dual; that xEQand xQE, and yEQ and yQE, are Hom E*-dual; and that there is
an isomorphism of rings
(6.5) E*(B(Zn+1 )) ~=U* [xE1; : :;:xEn]=((xEi)2 = xEixEi+1);
where i ranges over [n] and xEn+1is zero. In particular (6.5)applies to integral
cohomology, and completes the study begun at the start of the section. The
analogue of (6.4)is immediate.
We may substitute any doubly complex oriented spectrum D for E in
(6.5), on the understanding that left or right Chern classes must be chosen
consistently throughout. Duality, however, demands extra care and atten-
tion, and we take our cue from the universal example. We have to consider
the choice of splittings provided by Corollary 5.14, and the failure of formulae
such as (6.3)because the manifolds Bn are no longer double U-boundaries.
We are particularly interested in the left and right Chern classes xQ`, yQ`,
xQrand yQrin 2|Q|DU(Bn), and we seek economical geometric descriptions of
their Poincare duals. We continue to write xR and yR in DU2|R|(Bn) for the
homology classes represented by the respective inclusions of XR and YR with
their basic double U-structures.
Proposition 6.6. In DU2(n-|Q|)(Bn), we have that
0 Q0
Pd (xQ`) = yQ and Pd (yr ) = xQ ;
0 Q0
whilst Pd (xQr) and Pd (y` ) are represented by the inclusion of YQ and XQ
with the respective double U-structures
(YQ - ((jQ ) j*Qfl1)) ((jQ ) j*Qfl1) and (XQ - i*Qfl1) i*Qfl1;
for all n 0.
Proof. The first two formulae follow at once from Corollary 5.14, by analogy
with (1) of Theorem 6.2. The second two formulae require the interchange
of the left and right components of the normal bundles of jQ and iQ respec-_
tively. |__|
We extend to XQ in the obvious fashion.
Corresponding results for general D are immediate, so long as we continue
to insist that xDQand yDQin DDU2|Q|(Bn) are induced from the universal example
by the Thom class tD .
We apply Proposition 6.6 to compute the effect of the normal bundle map
` x r: XQ ! BU x BU
in *DU ( )-theory, for any Q [n]. By Corollary 5.11, this suffices to describe
`x r on our monomial basis_for G*. To ease computation, we consider an
alternative E*-basis (fiE) for E*(BU) (given any complex oriented spec-
trum E); this is defined as the image of the standard basis (fiE ) under the
homomorphism ?*, induced by the involution ?: BU ! BU of orthogonal
complementation. Since ?* acts by reciprocating the formal sum fiE , each
32 VICTOR M BUHSTABER AND NIGEL RAY
__ __ __
fiEnis an integral homogeneous polynomial in fiE1, : :,:fiEn. Moreover, ? is
a map of H-spaces with respect to Whitney sum, so that the relations
__E __E __E __E
(6.7) (fi1) 1(fi2) 2 : :(:fin) n = (fi )
__
continue to hold, and the elements fiEnagain form a polynomial basis. Since
?* is an involution, the dual basis _cEfor E*(BU) is obtained by applying_?*
to cE . We investigate the normal bundle map in terms of the basis fi fi!
for *DU (BU x BU ).
Fixing the subset Q = [jI(j) of [n], we consider the set H(Q) of non-
negative integer sequences h of the form (h1;P: :;:hn), where hi = 0 for any
i =2Q; for any such sequence h, we set |h| = ihi. Whenever h satisfies
P b(j)
i=lhi b(j) - l + 1 for all a(j) l b(j), we define the subset hQ Q
by
Xm
{m : hi = 0 for all a(j) l m b(j) };
i=l
otherwise, we set hQ = Q. For each j we write I(j) \ hQ as I(j; h), and
whenever I(j; h) is nonempty we denote its minimal element by m(j; h); we
then define subsets
A(h) = {m(j; h) : m(j; h) = a(j)} and M(h) = {m(j; h) : m(j; h) > a(j)}
of hQ. We identify the subset K(Q) H(Q) of sequences k for which ki
is nonzero only if i = a(j) for some 1 j s. Finally, it is convenient
to partition K(Q) and H(Q) into compatible blocks K(Q; ) and H(Q; )
for every indexing sequence ; each block consists of those sequences k or
h which have i entries i for each i 1, and all other entries zero. Thus,
for example, |h| = || for all h in H(Q; ). Any such block will be empty
whenever is incompatible with Q in the appropriate sense.
With this data, and for each k in K(Q) and h in H(kQ), we follow
the notation of Lemma 5.8 and write Xk+1hkQ;h[n]for the manifold XhkQ;h[n]
equipped with the double U-structure
i M M j
` = - fli ka(j)flm(j;h) and
khQ A(h)
M M
r = ka(j)+ 1 flm(j;h) fli;
A(h) M(h)
we note that m(j; h) = a(j) + ka(j)for each m(j; h) in A(h).
Theorem 6.8. When applied to the basic fundamental class in DU2|Q|(XQ ),
the normal bundle map yields
X i X X j __
g(k; h) fi fi! ;
;! K(Q;!) H(kQ;)
where the first summation ranges over all and ! such that || + |!| |Q|,
and g(k; h) 2 DU2(|Q|-|k|-|h|is represented by Xk+1hkQ;h[n]for all k 2 K(Q) and
h 2 H(kQ).
COBORDISM, FLAG MANIFOLDS, AND DOUBLES 33
__ __
Proof. We compute the coefficient of fih1: :f:ihn fik1: :f:ikn by repeatedly
applying Proposition 6.6, bearing in mind that the product structure in
*DU (Bn) allows us to replace any xmi (either left or right) by x[i;i+m-1]
when [i; i + m - 1] Q, and zero otherwise; indeed, the definitions of
H(Q) and K(Q) are tailored exactly to these relations. The computation
isPstraightforward,Palthough the bookkeeping demands caution, and yields
K(Q;!) H(kQ;_)g(k;_h)._We conclude by amalgamating the_coefficients of
those monomials fih1: :f:ihnand fik1: :f:ikn which give fi = and fi! respec-_
tively. |__|
ReadersLmay observe that our expression in x5 for ` as the sum of line
bundles_ ni=1iaei appears to circumvent the need to introduce the classes
fin. However, it contains n(n + 1)=2 summands rather than n, and their
Chern classes yi are algebraically more complicated than the xi used above,
by virtue of (6.1). These two factors conspire to make the alternative calcu-
lations less attractive, and it is an instructive exercise to reconcile the two
approaches in simple special cases. The apparent dependence of Theorem
6.8 on n is illusory (and solely for notational convenience), since ki and hi
are zero whenever i lies in Q0.
By combining Corollary 5.11 with (4.32) we may read off the values of
the double cobordism operations __s s! on the monomial basis for G*. The
__s
are occasionally referred to in the literature as tangential Landweber-
Novikov operations, and may be expressed in terms of the original s by
applying ?* and (6.7). We write
__ X
(6.9) fi = ; fi ;
where the ; are integers,PthePsummation ranges over sequences for
which | | = || and i i, and the equation holds good for both the
left and the right fis. We illustrate the procedure in the following important
special cases.
Corollary 6.10. Up to double U-cobordism, the actions of S*`and S*ron
monomial generators of G* are represented respectively by
X X X
s ;`(XQ ) = ; XhQ;h[n] and s!;r(XQ ) = Xk+1kQ;
H(Q;) K(Q;!)
wherePthePfirst summation ranges over sequences for which || = | | and
i i.
Proof. For s ;`(XQ ), we need the coefficient of fi 1 in Theorem 6.8. This
is obtained_by first_setting k = 0, so that kQ = Q, then collecting together
monomials fih1: :f:ihn 1 into the appropriate fi s and applying (6.9). For
s!;r(XQ ) we set h = 0, so that hQ = Q, and apply (6.9)in the corresponding_
fashion. |__|
We expect this result to provide a purely geometrical confirmation of
Corollary 4.29, that G* is closed under the action of S* S* on DU* ;
however, it remains to show that Xk+1kQlies in G*! We confirm that this is
the case after Proposition 6.15(2) below, but would prefer a more explicit
proof.
34 VICTOR M BUHSTABER AND NIGEL RAY
We may specialize Corollary 6.10 to the cases when and ! are of the
form ffl(k) for some integer 0 k |Q|, or when Q = [n] (so that we are
dealing with polynomial generators of G*), or both. We obtain
X b(j)-k+1X
(6.11) sffl(k);`(XQ ) = - YQ\[i;i+k-1]
j i=a(j)
X
and sffl(k);r(XQ ) = Xk+1Q\[a(j);a(j)+k-1];
j
where the summations range over those j for which b(j) - a(j) k - 1,
X X
(6.12) s ;`(X[n]) = ; Yh[n]
H([n];)
(
Xk+1[k+1;n]when ! = ffl(k)
and s!;r(X[n]) =
0 otherwise;
and
n-k+1X
(6.13) sffl(k);`(X[n]) = - Y[i-1][[i+k;n]:
i=1
These follow from the respective facts; ;ffl(k)= -1 when = ffl(k), and is
zero otherwise; K(Q; ffl(k)) consists solely of sequences containing a single
nonzero entry k in some position a(j); and K([n]; !) is empty unless ! = ffl(k)
for some 0 k n.
We may rewrite the proof of Theorem 6.8 by describing the duality in
more algebraic fashion. We suppose that g! is represented by the variety
XQ , where Q = [jI(j) and I(j) = [a(j); a(j)+t(j)-1] for some sequence of
integers (t(1);P: :;:t(s)) containing !i entries i for each i 1 (which requires
that s = i!i). We compute the image in DU2|Q|(BU x BU ) of the basic
fundamental class to be
X Ys
(6.14) gt(j)-(kj+hj;1+...+hj;t(j))(fi`)-1hj;1:(:f:i`)-1hj;t(j) *
*fikj;`;
kj;hj;1;:::;hj;t(j)j=1
where the summation ranges over all kj+ hj;1+ . .+.hj;t(j) t(j) such that
hj;m+. .+.hj;t(j) t(j)-m+1 for all 2 m t(j) and 1 j s. We may
convert to our preferred basis for DU* (BU x BU ) by using Lemma 3.11(3)
to express fikj;`in terms of the fin;r.
Before summarizing our conclusions, we consider two fascinating applica-
tions of the proofs of Theorem 6.8 and (6.14).
Proposition 6.15. We have that
(1)the map flh :Bn ! CP 1 represents either of the expressions
n+1-hX Xn n+1-hX
gn-k fik;` or gn-j(g)kj-kfik;r
k=0 k=0 j=0
in DU2n(CP 1 ), for each 1 h n;
COBORDISM, FLAG MANIFOLDS, AND DOUBLES 35
(2)the manifold X[k+1;n]represents (g)k+1n-kin DU2(n-k)when equipped with
the double U-structure
( - (k + 1)fl1) (k + 1)fl1;
for all 0 k n.
Proof. For (1), we note that the coefficient of fik;`in the first expression
is given by ; setting Q = [n] in (6.14)and concentrating on the
terms xkh;`, we obtain gn-k when 1 k n - h + 1, and zero otherwise, as
required. To convert the result into the second expression, we apply 3.11(3).
For (2), the second expression identifies (g)k+1n-kas Pd (xk1;r) in Bn. Con-
centrating on the terms xk1;rin Theorem 6.8 (or (6.12)), we deduce that this
__
is represented by Xk+1[k+1;n], as sought. |__|
The formulae in (1) reflect the fact that the map flh factors through the
(n - h + 1)-skeleton of CP 1 . Similar arguments for arbitrary Q generalize
Q kj+1
(2) to show that Xk+1kQrepresents j(g)t(j)-kj, which lies in in G* (as we
claimed after Corollary 6.10, and is also implicit in Theorem 6.16 below).
We conclude by summarizing the results that have motivated our entire
work, using the canonical isomorphism to identify G* and G* G* with S*
and S* S* respectively. We note that monomial generators of G* G*
may be expressed as double U-cobordism classes of pairs of basic double
U-manifolds (XQ ; XR ), for appropriate subsets Q and R [n].
Theorem 6.16. In the dual of the Landweber-Novikov algebra, the coprod-
uct ffi and antipode O are induced by the maps
X i X j
XQ 7! Xk+1kQ; XQ\kQ and XQ 7! O(XQ )
! K(Q;!)
up to double U-cobordism; similarly, in the quantum double D(S*), the com-
mutation law is induced by
X i X X j
s!XQ = ;!0Xk+1hkQ;h[n];
K(Q;!00)H(kQ;)
!0+!00=!
where ranges over those sequences for which ;!0 is nonzero.
Proof. For ffi, we apply Proposition 6.15(2) to Corollary 4.18, noting that
K(Q; !) is empty unless ! is compatible with Q; for O, we refer in addition
to Proposition 5.10. For the commutation law, we apply (6.9)to Theorem __
6.8 then appeal to the definition. |__|
Referring back to (6.13), we deduce the following special case of the com-
mutation law
n-k+1X
sffl(k)X[n]= - Y[i-1][[i+k;n]+ Xk+1[k+1;n]+ X[n]sffl(k)
i=1
for any k n and up to double U-cobordism. Taken with Corollary 6.10,
our results provide geometric confirmation of the formulae of (4.24) and
(4.31), once we have identified Y[i-1][[i+k;n]with bn-k and Xk+1[k+1;n]with
(b)k+1n-kunder the canonical isomorphism.
36 VICTOR M BUHSTABER AND NIGEL RAY
Intriguingly, we may represent the elements of DU* U DU* by threefold
*
U-manifolds, so that ffi is induced on a manifold M by modifying the double
U-structure from ` r to ` 0 r. The theory of multi U-cobordism is
especially interesting, and has applications to the study of iterated doubles
and the Adams-Novikov spectral sequence; we reserve our development of
these ideas for the future.
References
[1]J. Frank Adams. Stable Homotopy and Generalized Homology. Chicago Lectures *
*in
Mathematics. University of Chicago Press, 1974.
[2]J. Michael Boardman. The eightfold way to BP-operations. In Current Trends *
*in
Algebraic Topology, volume 2, Part 1 of CMS Conference Proceedings, pages 1*
*87-226.
American Mathematical Society, 1982.
[3]J. Michael Boardman. Stable operations in generalized cohomology. In I. M. *
*James,
editor, Handbook of Algebraic Topology, pages 585-686. North-Holland, Amste*
*rdam,
1995.
[4]Raoul Bott and Hans Samelson. Application of the theory of Morse to symmetr*
*ic
spaces. American Journal of Mathematics, 80:964-1029, 1958.
[5]Paul Bressler and Sam Evens. Schubert calculus in complex cobordism. Transa*
*ctions
of the American Mathematical Society, 331:799-813, 1992.
[6]V. M. Buchstaber and A. B. Shokurov. The Landweber-Novikov algebra and form*
*al
vector fields on the line. Funktsional Analiz i Prilozhen, 12:1-11, 1978.
[7]Victor M. Buchstaber. The Chern-Dold character in cobordism. Matematik Sbor*
*nik,
83:575-595, 1970.
[8]Victor M. Buchstaber. Semigroups of maps into groups, operator doubles, and*
* com-
plex cobordisms. In S. P. Novikov, editor, Topics in Topology and Mathemati*
*cal
Physics, volume 170 of American Mathematical Society Translations (2), page*
*s 9-
31, 1995.
[9]V. G. Drinfeld. Quantum groups. In Proceedings of the International Congres*
*s of
Mathematicians, Berkeley, volume 1, pages 789-820. American Mathematical So*
*ciety,
1987.
[10]A. D. Elmendorf. The Grassmannian geometry of spectra. Journal of Pure and *
*Ap-
plied Algebra, 54:37-94, 1988.
[11]Michiel Hazewinkel. Formal Groups and Applications. Academic Press, 1978.
[12]Howard H. Hiller. Geometry of Coxeter Groups. Number 54 in Research Notes in
Mathematics. Pitman, 1982.
[13]Toshiyuki Katsura, Yuji Shimizu, and Kenji Ueno. Complex cobordism ring and
conformal field theory over Z. Mathematische Annalen, 291:551-571, 1991.
[14]Ian G. Macdonald. Notes on Schubert Polynomials, volume 6 of Publications d*
*u Lab-
oratoire de Combinatoire et d'Informatique Mathematique. Universite du Queb*
*ec a
Montreal, 1991.
[15]Peter Magyar. Bott-Samelson varieties and configuration spaces. Preprint, N*
*ortheast-
ern University, 1996.
[16]J. Peter May, Frank Quinn, and Nigel Ray. E1 ring spectra. Springer Lecture*
* Notes
in Mathematics, 577:63-86, 1977.
[17]Susan Montgomery. Hopf Algebras and Their Actions on Rings, volume 82 of CB*
*MS
Regional Conference Series in Mathematics. American Mathematical Society, 1*
*993.
[18]Sergei P. Novikov. The methods of algebraic topology from the viewpoint of *
*cobordism
theories. Izvestiya Akademii Nauk SSSR, 31:855-951, 1967.
[19]Sergei P. Novikov. Various doubles of Hopf algebras: Operator algebras on q*
*uantum
groups, and complex cobordism. Uspekhi Akademii Nauk SSSR, 47:189-190, 1992.
[20]Daniel G. Quillen. On the formal group laws of unoriented and complex cobor*
*dism.
Bulletin of the American Mathematical Society, 75:1293-1298, 1969.
[21]Daniel G. Quillen. Elementary proofs of some results of cobordism theory us*
*ing Steen-
rod operations. Advances in Mathematics, 7:29-56, 1971.
COBORDISM, FLAG MANIFOLDS, AND DOUBLES 37
[22]Douglas C. Ravenel. Complex Cobordism and Stable Homotopy Groups of Spheres.
Academic Press, 1986.
[23]Nigel Ray. SU and Sp Bordism. PhD thesis, Manchester University, 1969.
[24]Nigel Ray. On a construction in bordism theory. Proceedings of the Edinburg*
*h Math-
ematical Society, 29:413-422, 1986.
[25]Nigel Ray and Robert Switzer. On SU xSU bordism. Oxford Quarterly Journal of
Mathematics, 21:137-150, 1970.
[26]Robert E. Stong. Notes on Cobordism Theory. Princeton University Press, 196*
*8.
[27]Moss E. Sweedler. Hopf Algebras. Mathematical Lecture Note Series. W. A. Be*
*njamin
Inc, 1969.
[28]Robert M. Switzer. Algebraic Topology, Homotopy and Homology, volume 212 of
Grundlehren der mathematischen Wissenschaften. Springer Verlag, 1976.
Department of Mathematics and Mechanics, Moscow State University,
119899 Moscow, Russia
E-mail address: buchstab@mech.math.msu.su
Department of Mathematics, University of Manchester,
Manchester M13 9PL, England
E-mail address: nige@ma.man.ac.uk
~~