COMPUTATIONS IN COMPLEX EQUIVARIANT BORDISM THEORY
DEV PRAKASH SINHA
1. Introduction
Bordism theory is fundamental in algebraic topology and its applications. In*
* the early
sixties Conner and Floyd introduced equivariant bordism as a powerful tool in t*
*he study of
transformation groups. In the late sixties, tom Dieck introduced homotopical b*
*ordism in
order to refine understanding of the localization techniques employed by Atiyah*
*, Segal and
Singer in index theory. Though equivariant bordism theories are fundamentally i*
*mportant
they have been mysterious from a computational point of view, even for cyclic g*
*roups of order
p where only a result by Kosniowski for geometric bordism [11] gives any explic*
*it information.
Recently, interest in equivariant bordism has grown in the wake of work on Mora*
*va K-theory
of classifying spaces by Hopkins, Kuhn and Ravenel [10], as a theorem of Greenl*
*ees and May
[9] shows that the homotopical equivariant bordism ring MUG*determines the stru*
*cture of
E-homology and cohomology of classifying spaces for any complex-oriented homolo*
*gy theory
E. Moreover, Kriz [12] has recent computations of MUZ=p*, which have quite a di*
*fferent form
from ours.
In this paper we present the first computations of the ring structure of the *
*coefficients of
equivariant bordism. We have a result which establishes an algebraic framework *
*in which to
understand equivariant bordism for any group such that any proper subgroup is c*
*ontained
in a proper normal subgroup. This class of groups includes abelain groups and *
*p-groups.
Our general result is computationally satisfying when one can find a suitable r*
*epresentation
of MUG*. For abelian groups, the map to completion at the augmentation ideal se*
*ems to be
such a representation. We state our results in the abelian case as follows.
Let ae denote the standard one-dimensional representation of S1. Let n denote*
* the repre-
sentation where S1 acts trivially on Cn. Choose as a model of the three-sphere *
*the set of all
pairs of complex numbers (z1; z2) such that |z1|2 + |z2|2 = 1.
Definition 1.1. Let P(V ) denote the (|V | - 1)-dimensional complex projective *
*space of lines
in V . In particular, P(n ae) is the projective space with an S1 action where *
*in homogeneous
coordinates i 2 S1 acts by multiplication on the last coordinate.
Definition 1.2. Define (X) for any stably complex S1-manifold to be the stably *
*complex
manifold __
(X) = X xS1S3 t (-X ) x P(1 ae);
__
where S3 has the standard Hopf S1-action, -X is the manifold obtained from X b*
*y imposing
a trivial action on X and taking the opposite orientation, and the S1-action on*
* X xS1S3 is
given by
i . [m; z1; z2]= [i . m; z1; iz2]:
1
2 DEV PRAKASH SINHA
Inductively define i(X) to be (i-1(X)), where 0(X) = X.
The importance of the construction is that it gives an explicit model by whi*
*ch we can
compute the image of a geometric class [X] (that is, a class in the image of th*
*e Pontrjagin-
1
Thom map) in the completion of MUS* at its augmentation ideal.
1
Theorem 1.1. Let [M] be class in MUS* which is the image under the Pontrjagin*
*-Thom
map of the class in geometric bordism represented by the complex S1-manifold M.*
* The
1
image of [M] under the map from MUS* to its completion at its augmentation idea*
*l, which
is isomorphic to MU*[[x]], is the power series
[ff(M)] + [ff((M))]x + [ff(2(M))]x2 + . .;.
where ff(i(M)) is the manifold obtained from i(M) simply by forgetting the G-ac*
*tion.
Definition 1.3. Let Yi(x) 2 MU*[[x]] be the image of the class [P(i ae)] under*
* the comple-
tion map.
We are now ready to state our main theorems. Let T = (S1)k. Though we will pr*
*ove the
following results from our techniques, we cite them now for convenience. Recall*
* (from [14]
for example) that for abelian groups G the completion of the equivariant bordis*
*m ring MUG*
at its augmentation ideal (which is the kernel of the forgetful map to ordinary*
* bordism)
is isomorphic to MU*(BG). For the torus, MU*(BT) is isomorphic to MU*[[x1; . .*
*x.k]].
Moreover, Landweber computed that more generally MU*(BG) is the quotient of MU**
*(BT)
by the ideal ([di]Fxi), where [d]Fx denotes the d-series of the universal forma*
*l group law over
MU* and the di are the orders of the cyclic factors of G when it is put in cano*
*nical form,
with di= 0 for a factor of S1.
Theorem 1.2. et G be an abelian group. If we embed G in a torus, the restric*
*tion map
MUT*! MUG*is surjective.
Theorem 1.3. The image of MUT*in its completion at the augmentation ideal is *
*contained
in the minimal sub-ring S of MU*[[x1; . .;.xk]] which satisfies the following t*
*wo properties:
o S contains the series Yi(f) where f ranges over [m1]Fx1+F . .+.F[mk]Fxk, Y*
*0(f) = f
and Yi(f) for i > 0 are defined above.
o If fff 2 S then ff 2 S, for f = [m1]Fx1 +F . .+.F[mk]Fxk.
We can recover the image of MUG*in its completion at the augmentation ideal f*
*or general
G by embedding G in a torus T and reducing the image of MUT*in MU*[[xi]] modulo*
* the
ideal ([di]Fxi) as above.
The structure of this paper is as follows. In the next two sections we review*
* basic defi-
nitions. In section 4 we establish the well-known connection between taking fix*
*ed sets and
localization in equivariant homotopy theory, in our setting of equivariant bord*
*ism. Section 5
is brief, introducing the algebraic language which has proven to be most useful*
* in equivariant
bordism theory. We prove our main theorems in section 6. We give geometric appl*
*ications in
section 7, in particular answering a question of Milgram about the geometry of *
*Lens spaces
and one of Bott about S1 actions on complex surfaces.
The author thanks his thesis advisor, Gunnar Carlsson, for pointing him to th*
*is problem
and for innumerable helpful comments. As this project has spanned a few years, *
*the author
COMPUTATIONS IN COMPLEX EQUIVARIANT BORDISM THEORY 3
has many people to thank for conversations which have been helpful including Bo*
*tvinnik,
Goodwillie, Klein, Milgram, Sadofsky, Scannell, Stevens and Weiss. Thanks also*
* go to
Greenlees, Kriz and May for sharing preprints of their work.
2.Prelimanaries
Until otherwise noted, the group G is a compact Lie group.
All G actions are assumed to be continuous, and G-actions on manifolds are as*
*sumed to
be smooth. For any G-space X, we let XG denote the subspace of X fixed under th*
*e action
of G. The space of maps between two G-spaces, which we denote Maps (X; Y ) has *
*a G-action
by conjugation. We denote its subspace of G-fixed maps by Maps G(X; Y ). Throug*
*hout, EG
will be a contractible space on which G is acting freely. And BG, the classifyi*
*ng space of G,
is the quotient of EG by the action of G.
We will always let V and W be finite-dimensional complex representations of *
*G. All
representations, and in fact all G-vector bundles, are assumed to carry a G-inv*
*ariant inner
product. Our G-vector bundles will always have paracompact base spaces. We wi*
*ll use
the same notation for a G-bundle over a point as for the corresponding represen*
*tation. Let
|V | denote the dimension of V as a complex vector space. The sphere SV is the *
*one-point
compactification of V , based at 0 if a base point is needed. And the sphere S(*
*V ) is the unit
sphere in V with inherited G-action. For a G-vector bundle E, let T (E) denote *
*its Thom
space, which is the cofiber of the unit sphere bundle of E included in the unit*
* disk bundle
of E. Thus for V a representation T (V ) = SV .
Let R+(G) denote the monoid (under direct sum) of isomorphism classes of comp*
*lex repre-
sentations of G, and let R(G) denote the associated Grothendieck ring (where mu*
*ltiplication
is given by tensor product). We let Irr(G) denote the set of isomorphism class*
*es of irre-
ducible complex representations of G, and let Irr*(G) be the subset of non-triv*
*ial irreducible
representations. Recall from the introduction that ae is the standard represent*
*ation of S1. We
will by abuse use ae to denote the standard representation restricted to any su*
*bgroup of S1.
We use n or Cn to denote the trivial n-dimensional complex representation of a *
*group. We
will sometimes think of representations as group homomorphisms, and talk of the*
*ir kernels,
images, and so forth. h i
For a ring R we let R 1_eidenote the localization of R in which ei2 R are in*
*verted.
3. Definition of MUG and Basic Properties Assumed
There are two basic definitions of bordism, geometric and homotopy theoretic.*
* Equivari-
antly, these two theories are not equivalent, and we will comment on this diffe*
*rence later in
this section.
Our main concern is the homotopy theoretic version of complex equivariant bor*
*dism, as
first defined by tom Dieck [7]. Fix U, a complex representation of G which con*
*tains all
irreducible representations of G infinitely often. If there is ambiguity possib*
*le we specify the
group by writing U(G). Let BUG (n) be the Grassmanian of complex n-dimensional *
*linear
subspaces of U. Let Gndenote the tautological complex n-plane bundle over BUG (*
*n). As in
4 DEV PRAKASH SINHA
the non-equivariant setting, the bundle Gnover BUG (n) serves as a model for th*
*e universal
complex n-plane bundle.
Definition 3.1. We let T UG be the pre-spectrum, indexed on all complex subrepr*
*esentations
of U, defined by taking the V th entry to be T (G|V)|(it suffices to define ent*
*ries of a prespectrum
only for complex representations). Define the bonding maps by noting that for V*
* W in U,
letting V ? denote the complement of V in W , we have
? G ? G
SV ^ T (|V)|~=T (V x |V)|:
Then use the classifying map
V ?x G|V!|G|W|
to define the corresponding map of Thom spaces. Pass to a spectrum in the usual*
* way, so
that the V th de-looping is given by
? G
limV (T (|W|));
WV
to obtain the homotopical equivariant bordism spectrum MUG .
We list some standard properties of MUG . For a thorough treatment of foundat*
*ions, see
the volume [14].
By construction, the homology theory which is represented by MUG has suspensi*
*on iso-
morphisms by any real representation of G. Such suspension isomorphisms are ne*
*cessary
for instance when constructing equivariant versions of transfer. The classifyin*
*g map of the
Whitney sum
G|Vx|G|W|! G|V W|
gives rise to a map
T (G|V)|^ T (G|W|) ! T (G|V W|);
which defines a multiplication on MUG . The unit element is represented by the*
* maps
SV ! T (G|V)|induced by passing to Thom spaces the classifying map of V viewed *
*as a G-
bundle over a point. Thus in the usual way the coefficients MUG*form a ring and*
* MUG*(X)
is a module over MUG*. Note that the coefficient rings of spectra which are ful*
*ly equivariant
(that is, indexed over U) are in general not commutative even in the graded sen*
*se. But MUG
exhibits a periodicity which simplifies matters.
Definition 3.2. Let V U be of dimension n. Then the classifying map V ! Gnindu*
*ces
a map of Thom spaces SV ! T (Gn), which represents an element tV 2 MUGV -2nknow*
*n as a
Thom class.
Proposition 3.1. The Thom class tV is invertible.
Proof.It is straightforward to show that the class represented by the map S2n !*
* T (G|V)|2
*
* __
MUG2n-Vinduced by the classifying map Cn ! G|Vi|s the multiplicative inverse of*
* tV. |__|
Corollary 3.2. The group MUGV(X) is naturally isomorphic to MUG2|V(|X).
COMPUTATIONS IN COMPLEX EQUIVARIANT BORDISM THEORY 5
The isomorphism is given by multiplication by tV. And this isomorphism is eas*
*ily extended
via suspension isomorphisms to real representations of G. Thus the grading of M*
*UG*, which
we have so far treated using subspaces of U, reduces to a Z-grading. And we hav*
*e chosen
specific compatible isomorphisms to reduce to this grading, namely multiplicati*
*on by Thom
classes.
Convention. We grade all MUG*-modules over the integers.
The most pleasant way to produce classes in MUG* is from equivariant stably a*
*lmost
complex manifolds. Recall that there is an real analog of BUG (n), which we cal*
*l BOG (n),
and which is the classifying space for all G-vector bundles.
Definition 3.3. A tangentially complex G-manifold is a pair (M; o) where M is a*
* smooth
G-manifold and o is a lift to BUG (n) of the map to BOG (2n) which classifies T*
* M x Rk for
some k.
We can define bordism equivalence in the usual way to get a geometric version*
* of equi-
variant bordism.
Definition 3.4. Let U;G*denote the ring of tangentially complex G-manifolds up *
*to bordism
equivalence.
Classes in geometric bordism give rise to classes in homotopical bordism thro*
*ugh the
Pontrjagin-Thom construction.
Definition 3.5. Define a map P T :U;G*! MUG*as follows. Choose a representative*
* M of
a bordism class. Embed M in some sphere SV , avoiding the basepoint and so that*
* the normal
bundle has a complex structure. Identify the normal bundle with a tubular neig*
*hborhood of
M in SV . Define P T ([M]) as the composite
T(f)
SV !c T () ! T (||);
where c is the collapse map which is the identity on and sends everything outs*
*ide to the
basepoint in T (), and T (f) is the map on Thom spaces given rise to by the cla*
*ssifying map
! ||.
The proof of the following theorem translates almost word-for-word from Thom'*
*s original
proof.
Theorem 3.3. The map P T is a well-defined graded ring homomorphism.
The Pontrjagin-Thom homomorphism is not an isomorphism equivariantly as it is*
* in the
ordinary setting, which is illustrated by the existence of the following classe*
*s in MUG*.
Definition 3.6. Compose the map SV ! T (Gn), in Definition 3.2 of the Thom clas*
*s with
the evident inclusion S0 ! SV to get an element eV 2 MUG-2nwhich is called the *
*Euler class
associated to V .
We will see that Euler classes eV associated to representations V such that V*
* G= {0} are
non-trivial. Thus MUG*is not connective, a feature which already distinguishes *
*it from U;G*
as well as MU* itself. The key difference between the equivariant and ordinary *
*settings is
6 DEV PRAKASH SINHA
the lack of transversality equivariantly. For example, if V G = {0} the inclusi*
*on of S0 into
SV cannot be deformed equivariantly to be transverse regular to 0 2 SV .
Finally, we introduce maps relating bordism rings for different groups. Recal*
*l that ordinary
homotopical bordism MU can be defined using Thom spaces as in our definition of*
* MUG
but without any group action present.
Definition 3.7. Define the augmentation map ff: MUG ! MU by forgetting the G-ac*
*tion
on MUG . When G is abelian and H is a subgroup of G define resH to be the map *
*from
MUG*! MUH* by restricting the G-action to an H-action.
The map resH is well-defined because any representation of H extends to a rep*
*resentation
of G so that the Thom space T (Gn) coincides with T (Hn) when its G-action is r*
*estricted to
an H-action.
Definition 3.8. Define the inclusion map : MU ! MUG by composing a map Sn ! T (*
*n)
with the inclusion T (n) ! T (Gn). More generally, we may define an inclusion f*
*rom MUG
to any MUGxH* by imposing a trivial H-action on a G-map from a sphere to Thom s*
*pace
and including T (Gn) into the T (GxHn) by including U(G) as the H-fixed set of *
*U(G x H)
and passing to Thom spaces.
On coefficients, defines an MU*-algebra structure on MUG*. The kernel of ff *
*on coeffi-
cients is called the augmentation ideal. For example, the Euler class eV is in *
*the augmentation
ideal as the map S0 ! SV in its definition is null-homotopic when forgetting th*
*e G-action.
Definition 3.9. Let x denote the image of x 2 MUG*under O ff.
Then the augmentation ideal contains, and is clearly generated by, elements o*
*f the form
x - x. On the other hand, is injective, which follows from the following propo*
*sition which
is proved for example in [14].
Proposition 3.4. The composite ff O : MU ! MU is homotopic to the identity map.
4. The Connection Between Taking Fixed Sets and Localization
Localization plays a central role in our computations. The connection between*
* localization,
in the commutative algebraic sense, and "taking fixed sets" has been a fruitful*
* theme in
equivariant topology. Computational work in equivariant bordism in the seventie*
*s, the last
time such work was seriously taken up, focused on construction of spectral sequ*
*ences based
on taking fixed sets. Our progress has arisen from taking the idea of localiza*
*tion more
seriously.
The following lemma provides translation between localization and topology.
Lemma 4.1. As rings, ^MUG*(S1 V ) ~=MUG*[_1_eV]:
Proof.Apply ^MUG* to the identification S1 V = lim-!SnV . After applying the *
*suspension
isomorphisms ^MUG*(SkV ) ~=MU^G*+|V(|Sk+1V ), the maps in the resulting direct*
*ed system
*
* __
are multiplication by the eV. *
* |__|
After inverting suitable Euler classes, the resulting localization is computa*
*ble.
COMPUTATIONS IN COMPLEX EQUIVARIANT BORDISM THEORY 7
Definition 4.1. A full set of representations of G is a set of representations *
*{Wi} such that
Wi G= {0} for all i but for any H G there is an i such that WiH 6= {0}.
Full sets of representations are useful because Z = Si(1 Wi) has fixed sets Z*
*G = S0 while
ZH is contractible for any H G. There are full sets of representations for any*
* group such
that any proper subgroup is contained in a proper normal subgroup. In particula*
*r, abelian
groups and p-groups have full sets of representations.
Theorem 4.2. Let {Wj} be a full set of representations of G. Then the localiz*
*ation of com-
plex G-equivariant bordism obtained by inverting the Euler classes eWj is a Lau*
*rent algebra
tensored with a polynomial algebra, given abstractly as follows:
ae oe
1 -1
MUG* ____ ~=MU* eV; eV ; Yi;V;
eWj
where V ranges over irreducible representations of G, i ranges over the positi*
*ve integers,
and where as indicated by notation eV is the image of the Euler class eV 2 MUG**
*under the
canonical map to the localization.
This theorem follows from a series of elementary, fairly standard lemmas. Our*
* first lemma,
taken with Lemma 4.1 and the existence of full sets of representations, establi*
*shes the strong
link between localization and taking fixed sets.
Lemma 4.3. Let X be a finite G-complex and let Z be a G-space such that ZG ' *
*S0 and
ZH is contractible for any proper subgroup of G. Then the restriction map
Maps G(X; Y ^ Z) ! Maps(XG ; Y G)
is a homotopy equivalence.
Proof.The fiber of this restriction map can be identified with the space of G-m*
*aps which are
trivial on XG , which is homotopy equivalent to the space of G-maps Maps G(X=XG*
* ; Y ^ Z).
Using the skeletal filtration of X=XG , we can then filter this mapping space b*
*y spaces
Mi= kMaps G(G=H +; Y ^ Z);
where H is a proper subgroup of G. A standard change-of-groups lemma says that *
*Mi is
homeomorphic to Maps (S0; (Y ^ Z)H ). But (Y ^ Z)H is contractible, and thus so*
* are the__
Mi, and thus so is the fiber of the restriction map. *
* |__|
We now translate this lemma to the stable realm. For simplicity, let us suppo*
*se that our
G-spectra are indexed over the real representation ring. We can do so by choosi*
*ng specific
representatives of isomorphism classes of representations. Let Kn Kn+1 denote *
*a sequence
of representations which eventually contain all irreducible representations inf*
*initely often
and such that Kn ? Kn+1 contains precisely one copy of the trivial representati*
*on. If G is
finite, we can let Kn be the direct sum of n copies of the regular representati*
*on.
Definition 4.2. Let X be a G-prespectrum. We define the geometric fixed sets s*
*pectrum
GX by passing from a prespectrum OEGX defined as follows. We let the entry {OEG*
*X}n be
(XKn)G, the G-fixed set of the Kn-entry of X. The bonding maps are composites
? G (K ?)G G G
(XKn)G- ! (Kn X(n+1)K) -! n (XKn+1) = (XKn+1) ;
8 DEV PRAKASH SINHA
where the first map is a restriction of a bonding map of X, and the second map *
*is restriction
to fixed sets of the loop space.
Lemma 4.4. Let Z be as in Lemma 4.3. Then for any G-prespectrum X, the presp*
*ectra
(X ^ Z)G and GX are homotopy equivalent.
Proof.Looking at the definition of (X ^ Z)G, consider
(W (XWV ^ Z))G:
G *
* G
Lemma 4.3 implies that restriction to fixed sets from this mapping space to W *
*(XWV )
G G
is a homotopy equivalence. Choosing V = Kn, we see that W (XWKn ) is an ent*
*ry of
OEGX. The bonding maps clearly commute with these restriction to fixed sets map*
*s, so_we_
have an equivalence of spectra. *
* |__|
Thus, the computation of MUG* localized at a full set of Euler classes reduce*
*s to the
computation of (GMUG )*. But for this latter computation we can use the geomet*
*ry of
Thom spaces. Because passing from prespectra to spectra commutes with taking s*
*mash
products (a consequence of uniquness of adjoints and the analogous statement fo*
*r function
spectra) we have
GMUG ~=Z ^ MUG ~=Z ^ T UG ~=GT UG ;
where we recall from Definition 3.1 that T UG denotes the equivariant Thom pres*
*pectrum.
We thus proceed with analysis of fixed-sets of Thom spaces.
We first need the following basic fact about equivariant vector bundles.
Proposition 4.5. And let E be a G-vector bundle over a base space with trivial *
*G-action
X. Then E decomposes as a direct sum
M
E ~= EV;
V 2Irr(G)
where EV ~=Ee V for some vector bundle eE.
Lemma 4.6 (tom Dieck). The G-fixed set of the Thom space of Gnis homotopy equi*
*valent
to 0 1
_ Y
T (|WG |) ^ @ BU()A ;
W2R+(G)n V 2Irr*(G) +
where R+(G)n is the subset of dimension n representations in R+(G) and *
*is the
coefficient of V in ff.
Proof.The universality of Gnimplies that (BUG (n))G is a classifying space for *
*n-dimensional
complex G-vector bundles over trivial G-spaces. Using Proposition 4.5 we see t*
*hat this
classifying space is 0 1
a Y
@ BU()A;
fi2R+(G) V 2Irr(G)
COMPUTATIONS IN COMPLEX EQUIVARIANT BORDISM THEORY 9
where is the coefficient of V in fi. Over each component of this union,*
* the universal
bundle decomposes as 1 x 2, where 1 is the universal vector bundle over the fac*
*tor of
BU(n) corresponding to the trivial representation. The fixed set of 2 G associ*
*ated to the
trivial representation is the entire bundle while the fixed set 2 G is the zero*
* section._ The
result now follows by passing to Thom spaces. *
* |__|
Theorem 4.2 now follows from the computation MU*(BU) ~=MU*[Yi], which is stan*
*dard
as in [1], where the Pontrjagin product using the multiplication on BU defines *
*the ring
structure on MU*(BU). Note that multiplication by Euler classes and their inver*
*ses serves
to change components.
We will also need the following geometric point of view, which dates back to *
*Conner and
Floyd. The following precise formulation is due to Costenoble.
Proposition 4.7. Let M be a tangentially complex G-manifold Then the normal bun*
*dle
of a connected component MG0of MG in M is a complex vector bundle.
Proof.Let j be a complex G-bundle over M whose underlying real bundle is T M x *
*Rk, as
given by the tangential unitary G-structure of M. Then by Proposition 4.5, j|MG*
*0decomposes
as a complex G-bundle
M
j|MG0~= j1 jae;
ae2Irr0(G)
where j1 has trivial G-action. But we can identifyLj1 as having underlying real*
* bundle equal
to T MG x Rk. So the normal bundle underlies ae2Irr0(G)jae, which gives th*
*e desired
*
* __
complex structure. *
* |__|
This proposition following would not be true if in the defintion of complex G*
*-manifold we
chose a complex structure on either the stable normal bundle or on T M x V for *
*an arbitrary
V as opposed to Rk. In these cases we could only guarantee that normal bundles *
*to fixed
sets would be stably complex.
Definition 4.3. Let
0 1
M Y
F* = MU*-|ff|@ BU()A
ff2R+(G) V 2Irr(G)
For a tangentially complex G-manifold M enumerate the connected components of M*
*G , and
label each component with the corresponding number as in MGi. Define the homomo*
*rphism
': U;G*! F* as sending a class [M] 2 U;Gnto the sum of the MGiwith reference ma*
*p to
the product of classifying spaces which classifies its normal bundle.
This geometric picture of taking fixed sets of G-actions on manifolds fits ni*
*cely with the
homotopy theoretic picture we have been developing so far.
10 DEV PRAKASH SINHA
Proposition 4.8 (tom Dieck). The following diagram commutes
'
U;G* ---! F*
? ?
? ?
yPT yi
MUG* ---! (GMUG )*:
We can explicitly compute '. A key fact which follows from standard knowledg*
*e of
MU*(BU) is the following.
Proposition 4.9. We can choose the class Yi;Vof Theorem 4.2 to be represented b*
*y the
image under the map i of PropositionQ4.8 of the complex projective space Pi wit*
*h reference
map to the V th factor in Irr*(G)BU which classifies the dual to the tautolog*
*ical line bundle.
Proposition 4.10. Let V behannirreducibleorepresentationiof G. The image of [P(*
*n V )]
in the localization MUG* 1__eWis Yn;V+ X, where X is (eV *)-n for one-dimensi*
*onal V
j
and is zero otherwise.
Proof.We use homogeneous coordinates. There are two possible components of the*
* fixed
sets. The points whose coordinates "in V " are zero, constitute a fixed Pn-1. *
* Alternately,
when all other coordinates are zero the resulting submanifold is the space of l*
*ines in V , which
is an isolated fixed point when V is one-dimensional and is a projective space *
*with no fixed_
points, as V has no non-trivial invariant subspaces, when V has higher dimensio*
*n. |__|
Definition 4.4. Let R0 be the sub-ring of MUG*generated by Euler classes and by*
* the classes
[P(n V )].
Corollary 4.11. The inclusion of R0 into MUG*induces an isomorphism after inver*
*ting the
Euler classes.
5.A Little Algebra
Definition 5.1. Let R0 be a sub-ring of a ring R. The saturation of R0 at a sub*
*set {ei} of
R is the smallest sub-ring S of R which contains R0 and such that for any ff 2 *
*R, if ffei2 S
then ff 2 S.
*
* _a0_
*
* ei0-a1
We may think of the saturation of R0 at {ei} as the ring of iterated fraction*
*s _____ei.,. .
*
* 1
where ai 2 R0. The saturation is canonically filtered as R0 R1 . .S., where R*
*j is the
minimal sub-ring containing Rj-1 and quotients of elements of Rj-1 by some ei.
Proposition 5.1. Let R0 be a sub-ring of a ring R such that the inclusion of R0*
* into R
becomes an isomorphism after inverting elements {ei} ofhR0,iand such that R0 co*
*ntains the
kernel of each of the canonical homomorphisms i:R ! R 1_ei. Then the saturatio*
*n of R0
at the {ei} is in fact all of R.
Proof.We first show thaththe saturationiof R0 at the {ei} contains the kernel o*
*f the canonical
homomorphism : R ! R {_1_ei}. Suppose x is in the kernel of so that xep1i1ep2*
*i2.e.p.kik= 0.
COMPUTATIONS IN COMPLEX EQUIVARIANT BORDISM THEORY 11
Then xep2i2.e.p.kikannihilates ep1i1and so is in the kernel of i1and thus R0. B*
*ut then x is in
the saturation of R0.
Now for any r 2 R, (r) = b=m where m is a monomial in the ei and by assumption
b 2 R0. Then r . m - b is in the kernel of and thus in R0 by the previous para*
*graph, so __
that m . r is in R0. Hence r is in the saturation of R0 at {ei}. *
* |__|
6. Computations of MUG*
Theorem 6.1. Let G be a group such that any proper subgroup is contained in a*
* proper
normal subgroup. Let R0 be the sub-ring of MUG* described above. The saturati*
*on at the
Euler classes of any sub-ring which contains R0 and the annihilators of Euler c*
*lasses is in
fact all of MUG*.
*
* __
Proof.Take Corollary 4.11 and Proposition 5.1 together. *
* |__|
Describing MUG* as the saturation of one of its sub-rings seems circular in r*
*easoning.
But we will prove facts about MUG*which make this description systematic. And a*
*s seen
through the map from MUG*to its completion at its augmentation ideal, this desc*
*ription as
a saturation works well and seems to be the simplest description possible.
We now focus on the case of abelian groups. We start by finding the annihilat*
*ors of Euler
classes.
Definition 6.1. Given a representation V let K(V ) denote its kernel, which is *
*the subgroup
of G which acts trivially on V .
Definition 6.2. For an abelian group G we let sG=H 2 MUG* denote the image of G*
*=H,
considered as a (framed, thus complex) G-manifold, under the Pontrjagin-Thom ma*
*p.
Theorem 6.2. Let V be a non-trivial irreduciblehrepresentationiof an abelian *
*group G. The
kernel of the localization map V :MUG* ! MUG* _1_eVis principal, generated by s*
*G=K(V )
which is zero if G=K(V ) is isomorphic to S1.
Proof.Apply ^MUG* to the cofiber sequence S(1 V )+ ! S0 ! S1 V . By Lemma 4.1, *
*the
second map of the sequence becomes the localization map V. Thus the kernel of V*
* is the
image of MUG*(S(1 V )) ! MUG*. We can begin to compute MUG*(S(1 V )). Because
S(1 V ) is fixed by K(V ) but then is a free G=K(V ) space we can use an Adams *
*transfer
to deduce that
MUG*(S(1 V )) ~=MUK(V*)-|G=K(V()|BG=K(V )):
As BG=K(V )here has a trivial K(V ) action there is an Atiyah-Hirzebruch spectr*
*al sequence
with
E2p;q~=Hp(BG=K(V ); MUK(Vq));
whose lim1term vanishes as Hp(BG=K(V )) is finite, converging to MUK(V*)(BG=K(V*
* )). Because
G=K(V ) is cyclic, its homology is generated by the unit class and images under*
* the classifying
map of fundamental classes of S(W )=G for any W which has a free action away fr*
*om zero. We
deduce that MUG*(S(1 V )) is generated as an MUG*-module by the image of the un*
*it class
and the spheres S(nV ). But the spheres S(nV ) bound the disks D(nV ), so the i*
*mage
12 DEV PRAKASH SINHA
is generated by the image of the unit class which is sG=K(V ). And if sG=K(V )i*
*s isomorphic to
*
* __
S1 then as a G-space it is S(V ) and thus is zero in MUG*. *
* |__|
Corollary 6.3. For a torus T the map from MUT* to its localization by inverting*
* Euler
classes is injective.
This corollary points to why localization methods have been so successful in *
*studying
S1-manifolds.
Remark. The analog of Theorem 6.2 is not true for geometric bordism for finit*
*e groups.
If G = Z=4 then sG is in the image of Ug;G*(S(1 ae2 )+) ! U;G*. Let H ~=Z=2 G.*
* In
MUG*, we have sG = sG=H . q where q is a quotient of eae2 by eae. But this clas*
*s q is not in
the image of the Pontrjagin-Thom map, and in fact sG is not divisible by sG=H i*
*n U;G*.
The following corollary is originally due to Comeza"na [14].
Corollary 6.4. For G abelian, MUG*is concentrated in even degrees.
Proof.Our original R0 and the kernels of these localizations are concentrated i*
*n even degrees.
When we divide by Euler classes to construct the saturation we continue to be c*
*oncentrated_
in even degrees. *
* |__|
For abelian groups, the restriction maps to subgroups are useful for organiza*
*tion.
Theorem 6.5. Let G be abelian, and let V be a representation of G. The kern*
*el of the
restriction map MUG*! MUK(V*)is principal, generated by eV.
Proof.Apply MUG *to the cofiber sequence
j V
S(V )+ !i S0 ! S :
The image of j is by definition the set of multiples of eV. To analyze the firs*
*t map we apply
MgU *Gto the composite of maps
k l 0
G=K(V )+ ,! S(V )+ ! S :
Since G=K(V ) and S(V ) are spaces which are fixed by K(V ) but are free G=K(V *
*) spaces,
the fact that the fixed-point spectra (MUG)H and (MUH )H are equivalent spectra*
* implies
* * * *
that gMU G(G=K(V )+) ~=gMU K(V()S0) and gMU G(S(V )+) ~=gMU K(V()S(V )=G+): Mor*
*eover,
the composite
* 0 * 0
l* O k*: gMUG (S ) ! gMU G=K(V()S )
coincides with the restriction map. But by Corollary 6.4,
* * 1 *
k*: gMUK(V )(S(V )=G+) = MUK(V )(S ) ! MUK(V )
is an isomorphism in even degrees. Hence, i* coincides with the augmentation ma*
*p in even_
degrees, which by Corollary 6.4 constitute all of MUG*. *
* |__|
COMPUTATIONS IN COMPLEX EQUIVARIANT BORDISM THEORY 13
Remark. There is a pleasing geometric construction which reflects the divisib*
*ility of classes
in the augmentation ideal by Euler classes. Let f :X ! Y be an S1-equivariant*
* map of
based spaces which is null-homotopic upon forgetting the S1 action. Let F :X x *
*I ! Y be
the null-homotopy. Construct an S1-equivariant map f(F) :X x I x S1 ! Y by send*
*ing
(x; t; i) 7! i . F (i-1 . x; t):
This map passes to the quotient
1 1 1
X x I x S1= {X x 0 x S } [ {X x 1 x S } [ {* x I x S };
which is Sae^ X. When restricted to S0 ^ X Sae^ X this map coincides with the *
*orginal
f, and thus gives a "quotient" of f by the class S0 ,! Sae.
Corollary 6.6. The augmentation ideal of MUG* is generated by classes {eVi} suc*
*h that
xVi 1 k
G ! (S ) is injective.
Proof.We may assume the inclusions of the kernels of Vi into G are split. Expre*
*ss x in the
augmentation ideal as
x = [x - r1(x)] + [r1(x) - r1 O r2(x))] + r1 O r2(x) - . .-.r1 O . .O.*
*rk(x);
where riis the map from MUG*to itself defined by composing the restriction map *
*to MUK(Vi)*
with the inclusion map of Definition 3.8. By assumption on the Vi, the composit*
*e r1O. .O.rk(x)
will be __xwhich is zero. And each difference y - ri(y) is in the kernel of the*
* restriction map
to K(Vi) and thus by Theorem 6.5 is divisible by eVi, yielding x as a linear co*
*mbination_of
the eVi. *
* |__|
We now reduce our study to the case in which G is a torus.
Theorem 6.7. Let H be a subgroup of an abelian group G. Then the restriction*
* map
MUG*! MUH* is surjective.
Proof.We use the characterization of MUH* as a saturation. To do so we must tak*
*e care in
choosing lifts of H-actions to G-actions. Factor the inclusion of H into G as H*
* ! eH! G
where eHis a subgroup of G which contains H, which is of the same rank as H, an*
*d whose
inclusion into G is split. We fix a splitting G ~=eHx F . Then to lift an H-rep*
*resentation to
a G-representation, we first lift it to a eH-representation and then to a G-rep*
*resentation by
having F act trivially.
Because representations lift, the H actions on projective spaces P(n V ) lif*
*t to G actions,
so the corresponding classes lift to MUG*. The classes sH=K(V )when zero-dimens*
*ional lift as
the quotient of eeVby any eW where eVis a lift of V to G as above and W is any *
*representation
with minimal kernel contained in the kernel of eV. Specifically, compose an equ*
*ivariant map
from SW to SVewith the unit map SVe! T (G|V)|to get a class which restricts to*
* sH=K(V ).
As MUH* is the saturation of the classes above at Euler classes, to prove the*
* theorem it
suffices to show that if a is divisible by an Euler class and a lifts to MUG*th*
*en its quotient
by that Euler class lifts to MUG*. Let us denote the lift of a to MUG*by ea. In*
*ductively, we
may assume that eais "acted on trivially" by F , by which we mean that the rest*
*riction of ea
to MUF*is in the image of the inclusion map from MU* to MUF*, as is the case fo*
*r liftings
14 DEV PRAKASH SINHA
of the classes above. That a is divisible by some eV is equivalent to, by Theor*
*em 6.5, the
restriction of a to MUK(V*)being zero. And since eais a lift of a its restricti*
*on to K(V ) is
zero. Moreover, as eais acted on trivially by F and the restriction of eato MU**
* MUK(V*)
is zero, earestricted to MUF* is zero. If we lift V to a G-representation eV *
*as above then
eV)
K(eV) = K(V ) x F so that earestricts to zero in MUK(* . Hence eais divisible *
*by eK(eV),
and the resulting quotient is a lift of the quotient of a by eV modulo some ann*
*ihilator of eV,
which we already have shown lifts. Finally, we note that the quotient of eaby e*
*eVwill_itself
be acted on trivially by F . *
* |__|
Definition 6.3. Fix a splitting sH of resH as a map of sets. And let oH = sH O *
*resH.
Definition 6.4. Impose an ordering on representations of T . Let be a sequen*
*ce of
representations = {V1; V2; . .V.m} such that Vi Vi+1 and let 0 = {V1; . .V.m-*
*1}. For
any x 2 MUT*inductively define x as the unique class such that eVm .x = x0 -o*
*K(Vm)(x0 ).
Theorem 6.8. As an MU*-algebra, MUT*is generated by classes P(n V ) and (eV*
*) , as
n ranges over positive integers, V ranges over irreducible representations of T*
* and ranges
over sequences of ordered representations as above.
Writing down relations among these generators is not enlightening because we *
*do not have
explicit understanding of the maps oH . So we focus instead on the map from MUT*
**to its
completion at its augmentation ideal.
Theorem 6.9. Let T = (S1)k. And given this decomposition let Vi be the repre*
*sentation
of T which restricts to ae, the standard representation, on the ith factor and *
*which restricts
to the trivial representation on the other factors. Let I be the augmentation i*
*deal of MUT*.
Then (MUT*)^Iis isomorphic to MU*[[xi]], the power series ring on i indetermina*
*tes, where
eVi7! xi.
The proof of this theorem is straightforward, as we have that the inclusion o*
*f MU* into
MUG* is split by the augmentation map and that for a torus T the annihilators o*
*f Euler
classes are zero. This theorem also follows from a completion theorem original*
*ly due to
L"offler and proved by Comeza"na and May in [14]. In fact, for abelian G comple*
*tion of MUG*
at its augmentation ideal coincides with the natural transformation of complex-*
*oriented
equivariant cohomology theories
MU*G(X) 7! MU*G(X ^ EG+ ) ~=MU*(X ^G EG +):
We are now positioned to understand the image of MUG*in its completion, which*
* has been
our goal.
1
Proposition 6.10. Choose the isomorphism (MUS*)^I~= MU*[[x]] where eae7! x as in
Theorem 6.9. The image of the Euler class eaen in this completion is [n]Fx, the*
* n-series in
the formal group law over MU*.
Proof.As the map from MUG*to its completion is a map of complex-oriented equiva*
*riant
cohomology theories, the Euler class of the bundle V over a point gets mapped t*
*o the Euler
class of V xG EG over BG. For G = S1, V = aen the resulting bundle is the nth-*
*tensor power __
of the tautological bundle over BS1, whose Euler class is by definition the n-s*
*eries. |__|
COMPUTATIONS IN COMPLEX EQUIVARIANT BORDISM THEORY 15
Remarkably, we can understand the image of geometric classes under completion*
* as well.
Recall Definition 1.2 of (X) for an S1 manifold X. We give an alternate descrip*
*tion now.
Definition 6.5. Let X be an S1 manifold. We let j(X) be_the_S1-equivariant bund*
*le over
P(1 ae) defined by taking the union of X x D(ae) with X x D(ae*) over their bo*
*undaries,
where the clutching function X x S1 ! X is given by the S1 action on X and the *
*stably
complex structure on the quotient is defined by using that on X and the standar*
*d complex
structure on P(1 ae).
__
Proposition 6.11. For any S1 manifold, (X) = j(X) - X x P(1 ae).
Note that if in the definition of j(X) we use not the standard complex struct*
*ure on P1
but the null-bordant complex structure, we have that (X) = j(X) without any cor*
*recting
term. This j-construction applied when X is a point gives rise to bounded flag *
*manifolds,
as studied extensively by Ray and his collaborators [15].
Theorem 6.12. For any complex S1-manifold X,
__
eae. [(X)] = [X] - [X ]
1
in MUS*.
*
* 1
Proof.By Corollary 6.3 it suffices to check the equality in a localization_of M*
*US* by inverting
Euler classes. By Proposition 4.8 we can compute the image of [X], [X ] and [(X*
*)] in the
localization at a full set of Euler classes by computing fixed sets with normal*
* bundle data.
The result_follows easily as the fixed sets of (X) are clearly those of X cross*
*ed with ae along
with an X crossed with ae. In the localization, crossing with ae coincides wi*
*th_dividing_by
eae. *
* |__|
1
Theorem 1.1, which states that the image of a geometric class [M] in MUS* ~=M*
*U*[[x]]
is
ff([M]) + ff([(M)])x + ff([2(M)])x2 + . .;.
follows as a corollary of Theorem 6.12.
And we can now also prove the our main theorem in the case of abelian groups *
*which we
can state now as follows. Recall that Yi(x) is defined to be the image of [P(i *
* ae)] under the
completion map to MU*[[x]], and Y0(x) = x.
Theorem 6.13 (Restatement of Theorem 1.3). The image of MUT*in its completion a*
*t the
augmentation ideal is contained in the saturation of MU*[Yi(f)] at the series f*
* 2 MU*[[x]],
where f ranges over [m1]Fx1 +F . .+.F[mk]Fxk.
Proof.First note that in general the image of the saturation of a sub-ring R0 *
*R at {ei}
under a homomorphism f is clearly contained in the saturation of f(R0) at {f(ei*
*)}. Given
Theorem 6.1 and Theorem 6.2 which together say that MUT*is the saturation of Eu*
*ler classes
and classes [P(iV )] at the Euler classes, along with the computations of these*
* classes under
completion in Proposition 6.10 and Theorem 1.1 we are almost done. It suffices*
* to check
that the image of [P(i aen ] is Yi([n]Fx), which follows from the fact that th*
*e S1 action on
[P(i aen ] is pulled back from the S1 action on [P(i ae)] by the degree n hom*
*omorphism_
from S1 to itself. *
* |__|
16 DEV PRAKASH SINHA
To recover the theorem for all abelian groups we cite Theorem 6.7 and the fac*
*t that after
completion, restricting to a subgroup K(V ) coincides with taking the quotient *
*of the power
series ring modulo the image under completion of eV.
Remark. The completion at the augmentation ideal is a natural place in which *
*to view
1
MUG*as a saturation. Suppose f = a0 + a1x + a2x2 + . .i.s in the image of MUS* *
*under
completion. Then because R0 contains MU* we have that f - a0 = a1x + . .i.s in *
*the image
of the saturation. But the saturation property at x = ^I(eae) implies that a1+ *
*a2x + a3x2+ . . .
is in the image. More generally, any ai+ ai+1x + . .i.s in the image. So the *
*property of
a series being in the image of the completion map depends only on the tail of t*
*he series.
Perhaps there is an analytic way to define this image.
A similar statement to this theorem holds for K-theory, again for abelian gro*
*ups. For
example, K*S1~=Z[ae; ae-1]. If we complete at the augmentation ideal, a generat*
*or for which
we choose the Euler class ae - 1, we see that the image is generated over the i*
*ntegers by x,
which is the 1-series and by -x + x2 - x3 + . .w.hich is the -1-series in the m*
*ultiplicative
formal group law. In fact all the n-series are in this image, but they are alre*
*ady in the sub-
ring generated by the 1 and -1 series as the multiplicative formal group law is*
* a polynomial.
Moreover, the image is saturated, but once again one does not produce new class*
*es in this
way.
Because the completion at the augmention ideal is a natural place in which to*
* understand
equivariant bordism, we hope that the map from MUG*to this completion is inject*
*ive. We
also hope to understand whether this image is all of the saturation in the comp*
*letion or
whether the containment is proper. These questions are linked.
Definition 6.6. We say that a homomorphism f :R ! B is clean (with respect to {*
*ei}) if
the image of the saturation of R0 at {ei} is the saturation of the image of R0 *
*at the image
of {ei}
Theorem 6.14. For G = Z=p the homomorphism from MUG*to its completion at its *
*aug-
mentation ideal MU*[[x]]=([p]Fx) is clean with respect to Euler classes.
Proof.The following diagram commutes
^I
MUG* ---! MU*[[x]]=([p]Fx)
? ?
? ?
yff yf7!f(0)
MU* --=-! MU*:
If ^I(a) is divisible by some [n]Fx in the completion for some n < p then it ma*
*ps to 0 in
MU*. So ff(a) = 0. Corollary 6.6 implies that a is divisible by eaen, which is *
*what_was to
be shown. *
*|__|
For general abelian groups such a proof does not work because for we do not k*
*now that
the map MUK(V*)! (MUK(V*))^Iis injective except for when K(V ) is the trivial g*
*roup. In
fact, the completion map MUG*! (MUG*)^Iis clean with respect to Euler classes i*
*f and only
if this completion map is injective for all subgroups of G.
There are alternate representations of MUG*for G abelian which are clean and *
*injective.
COMPUTATIONS IN COMPLEX EQUIVARIANT BORDISM THEORY 17
h i
Definition 6.7. Let H :MUG* ! MUH* _1_eWbe the composition of the restriction m*
*ap
i
from MUG*to MUH* and the canonical map from MUH* to its localization by inverti*
*ng a full
set of Euler classes.
Theorem 6.15. The map
Y Y 1
H :MUG*! MUH* ___
HG HG eWi
is injective for G abelian.
Proof.We introduce the localization filtration. Consider
1 G 1 1
MUG*! MUG* ___ ! MU* ___; ___ ! . .;.
eV1 eV1 eV2
where the Vi are non-trivial irreducible representations. As in Theorem 6.2, we*
* can fit the
map
1 G 1 1
MUG* ___ ! MU* ___; ___
eVi eVi eW
h i
into an exact sequence whose third term is MUK(W)*-11_eV(BG=K(W)) and compute t*
*he kernels
i
of the localizationhmapsias generated by sG=K(W). But then these kernels map i*
*njectively
to MUK(W)*-11_eV, from which by induction in the filtration we can prove that M*
*UG*maps
i *
* __
injectively to their product. *
* |__|
By Proposition 4.8, this theorem says that we have a good understanding of eq*
*uivariant
bordism theory by restricting to subgroups and then looking at fixed-sets with *
*normal bundle
data. In fact this point of view was our starting point. But it is difficult to*
* discern for any
given fixed-set data whether is the fixed-set data of a G manifold, or from the*
* localization
point of view whether fixed-set data is integral. The saturation has proven to *
*be a way to
generate all integral fixed-set data given a relatively small amount of such da*
*ta with which
to begin. Unfortunately, weQdo not know how to compute the image of the saturat*
*ion of R0
at the Euler classes under H to get a faithful description of MUG*even though*
* we know
how to compute the images of the classes in R0 themselves.
In the context of looking at fixed sets, there is a geometric interpretation *
*of the canon-
ical filtration of MUG*arising from its construction as a saturation. For geome*
*tric classes
[M] in MUG*define the depth of isotropy to be the maximum length of a chain of *
*proper
containments MH1 MH2 . . .M, taking the minimum over bordism representatives
to get a well-defined number. The Ri-1in which a geometric class in MUG*first *
*appears
corresponds roughly to the depth of its isotropy. All [P(i V )] have depth 1 a*
*s there are no
subgroups H such that P(i V )G P(i V )H P(i V ). But for example, the quot*
*ient
1 S1 Z=2
[V ] of [P(1 ae)] - [P(1 ae3 )] by eae2in MUS* will have V V V .
18 DEV PRAKASH SINHA
7. Applications and Further Remarks
In this section we give an assortment of applications and indicate directions*
* for further
inquiry.
Our first application answers a question about bordism of free Z=n-manifolds *
*posed to us
by Milgram. It is well-known that the spheres S(kaem ) for any m relatively p*
*rime to n
generate MU*(BZ=n) as an MU*-module. How are these bases related?
Theorem 7.1. Let m and n be relatively prime. Let Q(x) be a quotient of x by*
* [m]Fx
modulo [n]Fx in MU*[[x]]. Define ai2 MU* by (Q(x))k = a0 + a1x + a2x2 + . ...Th*
*en
[S(kaem )] = a0[S(kae)] + a1[S(k-1ae)] + . .+.ak-1[S(ae)]
in MU*(BZ=n).
Proof.We use an analog of the simple fact that if M is a G-manfold and M r MG h*
*as a
free G-action then [@(MG )] = 0 in MU*(BG), where @(MG ) is the boundary of a t*
*ubular
neighborhood around the fixed set MG . The null-bordism is defined by M r (MG *
*). If
the fixed points of M are isolated, this will give rise to a relation among sph*
*eres with free
G-actions.
Let ff0 = qk where q is a quotient of eaeby eaem in MUZ=n*. Inductively, let *
*ffibe a quotient
of ffi-1- ____ffi-1by eae. Then the "fixed sets" of ffk are given by
(ffk) = eaem-k- ___ff0eae-k- ___ff1eae-k+1- . .-.____ffk-1eae-1:
As eV-1 corresponds to a tubular neighborhood of an isolated fixed point in geo*
*metric
bordism, we can deduce via transversality arguments for free G-actions that
[S(kaem )] - ___ff0[S(kae)] - ___ff1[S(k-1ae)] - . .-.____ffk-1[S(*
*ae)] = 0
in MU*(BZ=n). But the image of ff0 in (MUZ=n*)^I~=MU*[[x]]=[n]Fx is (Q(x))k fro*
*m which
we can read off that __ffi= ai.
Note that our expressions in MU*(BZ=n) are independent of the indeterminacy i*
*n choosing_
q and the ffi. *
* |__|
This old idea of using G-manifolds to bound and thus give insight into free G*
*-manifolds has
been codified by Greenlees's introduction of local cohomology to equivariant st*
*able homotopy
theory [8]. Essentially, MU*(BG) is isomorphic to the cohomology of the Koszul *
*complex
O 1
MUG*! MUG* ___
V 2B eV
Q
where S is a set of representations of G such that G acts freely on V 2BS(V )*
*. We have
reproduced known Tor classes in MU*(B(Z=2)k) using this method. Results along t*
*hese lines
will appear in [17]
Our next application is in answer to a question posed to us by Bott. For sim*
*plicity,
define the fp-signature of an isolatedLfixed point in an S1 manifold whose norm*
*al bundle is
isomorphic to the representation ki=1aemi to be the unordered k-tuple {m1; .*
* .;.mk}. The
fp-signature of an S1 action with isolated fixed points is the unordered set of*
* fp-signatures
of the fixed points. For example, the fp-signature of P[1 aem aen ] is {{m; *
*n}; {-m; n -
COMPUTATIONS IN COMPLEX EQUIVARIANT BORDISM THEORY 19
m}; {m - n; -n}}. Bott's question is as to whether there are any other fp-signa*
*tures which
arise from S1 actions on CP2 with three isolated fixed points. In fact, a surfa*
*ce admitting
an S1 action with three isolated fixed points is a surprisingly restrictive con*
*dition.
Theorem 7.2. Let M be a complex four (real)-dimensional S1-manifold with thre*
*e isolated
fixed points. Then the fp-signature of M is {{m; n}; {-m; n - m}; {m - n; -n}} *
*for some
integers m and n. Moreover, forgetting the S1 action M is bordant to CP2.
1 *
* 1
Proof.For convenience, let refer to the Euler class eaen 2 MUS* by en. A compl*
*ex S
1
manifold M with three isolated fixed points defines a class in MUS* whose image*
* under is
([M]) = e-1ae-1b+ e-1ce-1d+ e-1fe-1g
for some integers a; . .;.g. We let T denote
1
ecedefeg + eaebefeg + eaebeced 2 MUS*:
And we say that a class in a localization is integral when it is in the image o*
*f the canonical
map to that localization.
Without loss of generality, assume a is greatest of the integers a; . .;.g in*
* absolute value.
As T must be divisible by ea, Theorem 6.5 implies that T restricted to MUZ=a*mu*
*st be zero.
The Euler class en restrict non-trivially to MUZ=a*unless a|n. Therefore one of*
* c; d; f; g, say
c must behequalito a. We first claim that this number must be -a. Look in the l*
*ocalization
1 e1
of MUS* {__ea}defined by inverting all Euler classes except for ea This locali*
*zation is itself
a saturation, in this case only at ea. Suppose that |b|; |d|; |f|; |g| < |a| an*
*d that
e-1ae-1b+ e-1ae-1d+ e-1fe-1g
is integral, by which we mean in the image of this partial localizationhmap.iTh*
*en we must
have that e-1b+ e-1ddivisible by ea and thus is zero in MUZ=a*_1_eV, which is i*
*mpossible.
Cases where some of |b|; |d|; |f|; |g| are equal to |a| can be treated similarl*
*y.
As c = -a and again assuminghthat |b|;i|d|; |f|; |g| < |a|, consider the clas*
*s ea([M]) -
1 e1
ea([P(1 + aea )])e-1din MUS* {__ea}. Its image under full localization
e-1ae-1b- e-1ae-1d+ e-1fe-1g;
1 h e1 i
which implies that e-1b- e-1dis divisible by ea in MUS* {__ea}or that b d (mo*
*d a). But
because |b|; |d| < |a| we have that d = a b depending on whether b is positive*
* or negative.
We can once again eliminate cases where some |n| = |a|.
Finally, as c = -a and d = b - a consider ([M] - [P(1 aea aeb )]), which w*
*ill be equal
e-1fe-1g- e-1a-be-1-b. Case analysis of necessary divisibilities as we have bee*
*n doing implies that
this difference must be zero.
Finally, by Theorem 6.2 this fixed-set data determines [M] as in S1-equivaria*
*nt homotopi-
1
cal bordism uniquely, so that [M] must equal [P[1 aem aen ]] in MUS4. Forget*
*ting the
S1 action, this equality gives rise to a bordism between [M] and [CP2]. *
* __
*
*|__|
20 DEV PRAKASH SINHA
In fact, we have show that any complex four-dimensional S1-manifold M with th*
*ree fixed
points is "stably cobordant" to one of the standard actions on P2. The question*
* of whether
this equivalence gives rise to an actual equivariant bordism between M and such*
* a P2 reduces
to a transversality question. In fact, transversality questions which arise in *
*real equivariant
bordism are universal in a sense [6]. Hence, our techniques could be useful in *
*studies which
require control of the failure of equivariant transversality.
Equivariant bordism also underlies the study of equivariant genera. Our work *
*illuminates
known results and could lead to further developments. Ochanine proved that any *
*(strongly)
multiplicative genus, which is a genus whose value on the total space of a fibe*
*r bundle of
manifolds is the product of the genus of the base and fiber, is rigid in the se*
*nse that the
equivariant genus obtained using the Borel construction and "integration along *
*the fiber" will
not depend on the G-action on the manifold (see for example [16]). But such an *
*equivariant
genus factors through completion of equivariant bordism at its augmentation ide*
*al, and
Theorem 1.1 show that coefficients of terms other than the constant term in the*
* image
under completion will be differences of twisted bundles over P1 and trivial bun*
*ldes over P1.
Any (strongly) multiplicative genus will thus vanish on these terms.
Finally, we reiterate that the theories we have been studying provide a unifi*
*ed framework
in which to study the characteristic classes E*(BG) for any complex-oriented th*
*eory E. We
hope that our understanding of relevant commutative algebra can lead to new ins*
*ights into
these characteristic classes.
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