COMPUTATIONS OF COMPLEX EQUIVARIANT BORDISM
RINGS
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 the study of transformation groups. In the late sixties, tom Dieck introduc*
*ed
homotopical bordism in order to refine understanding of the localization techni*
*ques
employed by Atiyah, Segal and Singer in index theory. Despite the many successf*
*ul
computations and applications of bordism theories, equivariant bordism has been
mysterious from a computational point of view, even for cyclic groups of prime
order p (see [11] and [12]). In this paper we present the first computations of*
* the
ring structure of the coefficients of equivariant bordism, for abelian groups. *
* The
key constructions are operations on equivariant bordism. Analogs of these opera-
tions should play an important role in equivariant stable homotopy more general*
*ly.
Our main techniques involve localization and give some insight into the structu*
*re
of MUG*for a large class of groups including p-groups. We give a synopsis of our
results now.
We denote by MUG* the homotopical equivariant bordism ring, where G is a
compact Lie group. It is defined analogously to MU* as limV[Sn_V ; T (G|V)|]G ,
where V ranges over isomorphism classes of complex representations of G, Sn_V
is the one-point compactification of the Whitney sum of Cn with trivial G action
and V , and T (G|V)|is the Thom space of the universal complex G-bundle. In fac*
*t,
we may use these Thom spaces to define an equivariant spectrum as first done by
tom Dieck [7] and hence define associated equivariant homology and cohomology
theories MUG*(-) and MU*G(-). We will carefully make these constructions in
section 3.
Euler classes play fundamental roles in our work. The Euler classes which are
most important for us are those associated to a complex representation of G, co*
*n-
sidered as a G-bundle over a point. More explicitly, the Euler class associated*
* to V
is a class eV 2 MUmG(pt:), where m is the dimension of V over the reals, repres*
*ented
by the composite S0 ,! SV ! T (G|V)|, where the second map is "inclusion of a
fiber". Euler classes multiply by the rule eV . eW = eV W . In homological gra*
*ding
eV 2 MUG-m, so it cannot be in the image of a geometric bordism class under the
Pontrijagin-Thom map if it is non-trivial. If V G = {0} then eV is non-zero, re*
*flect-
ing the fact that V has no non-zero equivariant sections. Therefore, the homoto*
*py
groups of MUG are not bounded below, a feature which already distinguishes it
from its ordinary counterpart.
More familiar classes in MUG* are those in the image of classes in geometric
bordism under the Pontrijagin-Thom map. Given a stably complex G-manifold M,
1
2 DEV PRAKASH SINHA
let [M] denote the corresponding class in MUG*. Complex projective spaces give
a rich collection of examples of G-manifolds. Given a complex representation W
of G let P(W ) denote the space of complex one-dimensional subspaces of W with
inherited G-action.
The starting point in our work is that after inverting Euler classes, MUG*bec*
*omes
computable by non-equivariant means. That we rely heavily on localization is not
surprising because localization techniques have pervaded equivariant topology. *
*For
any compact Lie group G let R0 denote the sub-algebra of MUG*generated by the
eV and [P(n_V )] as V ranges over non-trivial irreducible representations. Let *
*S be
the multiplicative set in R0 of non-trivial Euler classes. By abuse, denote the*
* same
multiplicative set in MUG*by S. Then the key first result, which we emphasize is
true for a large class of groups including p-groups, is the following.
Theorem 1.1. Let G be a group such that any proper subgroup is contained in a
proper normal subgroup. The inclusion of R0 into MUG*becomes an isomorphism
after inverting S.
In other words, we may multiply any class in MUG*by some Euler class to get
a class in R0 modulo the kernel of the localization map S. We are lead to study
divisibility by Euler classes as well as the kernel of this localization map. W*
*e can
do so successfully in the case when the group in question is a torus.
Let T be a torus, and let V be a non-trivial irreducible representation of T *
*. Let
K(V ) denote the subgroup of T which acts trivially on V . There is a restricti*
*on ho-
morphism (of algebras) resTH:MUT*! MUH*for any subgroup H. The restriction
of eV to MUK(V*)is zero, as can be seen using an explicit homotopy. Remarkably,
we have the following.
Theorem 1.2. The sequence
resTK(V )K(V )
0 ! MUT*.eV!MUT* ! MU* ! 0
is exact.
Note that the surjectivity of the restriction map is false for geometric bord*
*ism.
One cannot for example extend the non-trivial action of Z=2 on two points to an
S1-action.
Using this exact sequence, we define operations which are essentially divisio*
*n by
Euler classes. To define these operations we need to split the restriction maps*
*. The
restriction map to the trivial group is called the augmentation map ff: MUG* !
MU*. There is a canonical splitting of this map as rings which defines an MU*-
algebra structure on MUG*. All of the maps we have defined so far are in fact m*
*aps
of MU*-modules. The restriction maps to other sub-groups are not canonically
split, but we do know the following from [14].
Theorem 1.3 (Comeza"na). Let G be abelian. Then MUG* is a free MU*-module
concentrated in even degrees.
Hence we may fix a splitting sV as MU*-modules of the restriction map resTK(V*
*.)
Unless K(V ) is the trivial group, this splitting is non-canonical and is not a*
* ring
homomorphism.
Definition 1.4.Let T and V be as above. Define the MU*-linear operation V as
follows. Let x 2 MUT*. Then V (x) is the unique class in MUT*which satisfies
eV . V (x) = x - sV (resTK(Vx)):
COMPUTATIONS OF COMPLEX EQUIVARIANT BORDISM RINGS 3
For convenience, let fiV denote sV O resTK(V.) We are now ready to state our
main theorem.
Theorem 1.5. For any choice of splittings sV , MUT* is generated as an MU*-
algebra over the operations V by the classes eV and [P(n_ V )], where n 2 N and
in all instances V ranges over non-trivial irreducible complex representations *
*of T .
Relations are as follows:
o eV V (x) = x - fiV (x)
o V (fiV (x)) = 0
o V (eV ) = 1
o V (xy) = V (x)y + fiV (x)V (y) + V (fiV (x)fiV (y))
o V W x = W V x + W V fiW (x) - W V (eW )fiV (W x)
- W V (fiV (eW )fiV (W x));
where V and W range over non-trivial irrreducible representations of T and x and
y are any classes in MUG*.
We may recover the structure of MUG*for any abelian group G by realizing G
as the kernel of an irreducible representation of some torus and using the exact
sequence of Theorem 1.2.
We give both algebraic and geometric applications of our main computation.
For G = S1 and ae its standard representation, we present a geometric model of
ae([M]). This geometric model allows us to compute the completion map MUG*!
(MUG*)^I, where I is the kernel of the augmentation map from MUG* to MU*.
The completion theorem of L"offler, as proved by Comeza"na and May, states that
for G abelian, (MUG*)^I~= MU*(BG ), where BG is the classifying space of G.
So this completion map gives a connection between equivariant bordism and any
equivariant theory which is defined using a Borel construction EG xG -. We also
give more classically-styled applications to the understanding of group actions*
* on
manifolds. For example, a current topic of great interest in equivariant cohomo*
*logy
is the investigation of G-manifolds with isolated fixed points, essentially ext*
*ending
Smith theory. We prove the following.
Theorem 1.6. Let M be a stably-complex four dimensional S1-manifold with three
isolated fixed points. Then M is equivariantly cobordant to P(1_ V W ) for some
distinct non-trivial irreducible representations V and W of S1.
The author thanks his thesis advisor, Gunnar Carlsson, for pointing him to
this problem and for innumerable helpful comments. As this project has spanned
a few years, the author has many people to thank for conversations which have
been helpful including Botvinnik, Goodwillie, Klein, Milgram, Sadofsky, Scannel*
*l,
Stevens and Weiss. He also thanks Haynes Miller for a close reading of an earli*
*er
version of this paper. Thanks also go to Greenlees, Kriz and May for sharing
preprints of their work.
2.Preliminaries
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
assumed to be smooth. For any G-space X, we let XG denote the subspace of
X fixed under the 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
4 DEV PRAKASH SINHA
G-fixed maps by Maps G(X; Y ). We will often work with based spaces, in which
case we assume that the basepoints are fixed by G. Throughout, EG will be a
contractible space on which G is acting freely. And BG , the classifying 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.
Our G-vector bundles will always have paracompact base spaces, so we may define
a G-invariant inner product on the fibers. The constructions we make using such
an inner product will be independent of choice of inner product up to homotopy.
We will use the same notation for a G-bundle over a point as for the correspond*
*ing
representation. We 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 poi*
*nt
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
complex representations of G, and let R(G) denote the associated Grothendieck
ring (where multiplication is given by tensor product). We let Irr(G) denote th*
*e set
of isomorphism classes of irreducible complex representations of G,Pand let Irr*
**(G)
be the subset of non-trivial irreducible representations. If W = aiVi 2 R(G)
where Viare distinct irreducible representations, let V (W ) for an irreducible*
* V be
aj if V is isomorphic to Vj or zero if V is not isomorphic to any of the Vi. Re*
*call
from the introduction that ae is the standard representation of S1. We will by *
*abuse
use ae to denote the standard representation restricted to any subgroup of S1. *
*We
use n_or Cn to denote the trivial n-dimensional complex representation of a gro*
*up.
We will sometimes think of representations as group homomorphisms, and talk of
their kernels, images, and so forth.
We rely on techniques from equivariant stable homotopy theory. Let W (X)
denote the space of based maps from SW to X. By fixing a representation U with
inner product, of which a countably infinite direct sum of any representation o*
*f G
appears as a summand, we define a G-spectrum X to be a family of spaces XV
indexed on subspaces of U equipped with G-homeomorphisms XV ! WV XW
for all V W . The basic passage to ordinary stable homotopy theory is by taking
the fixed-points spectrum. Consider only subspaces V UG . Then we may define
the fixed-points spectrum XG using the family of spaces (XV )G , where the bond*
*ing
maps are restrictions to fixed sets of the given bonding maps.
3.Basic Properties of MUG
There are two basic definitions of bordism, geometric and homotopy theoretic.
Equivariantly, these two theories are not equivalent, and we will comment on th*
*is
difference 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 whi*
*ch a
countably infinite direct sum of any representation of G appears as a summand. *
*If
there is ambiguity possible 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 the non-equivariant
COMPUTATIONS OF COMPLEX EQUIVARIANT BORDISM RINGS 5
setting, the bundle Gnover BUG (n) serves as a model for the universal complex
n-plane bundle. If V is a complex representation, set GV= G|V |.
Definition 3.1.We let T UG be the pre-spectrum, indexed on all complex subrep-
resentations of U, defined by taking the V th entry to be T (GV) (it suffices t*
*o define
entries 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 ?
SV ^ T (GV) ~=T (V ? x GV):
Then use the classifying map
V ?x GV! GW
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
limWVV (T (W ));
to obtain the homotopical equivariant bordism spectrum MUG .
From this spectrum indexed by subspaces of U we may pass to an RO(G)-graded
homology theory MUG?(-). We will be concerned with the coefficient ring in inte*
*ger
gradings, which we denote MUG*. But for some arguments, we will need groups
graded by complex representations of G, giving rise to the need for the followi*
*ng
proposition.
Proposition 3.2.Let V be a complex representation of G. The group MUGV(X)
is naturally isomorphic to MUG2|V(|X).
We prove this proposition after defining the needed multiplicative structure *
*on
MUG . The classifying map of the Whitney sum
GVx GW ! GV W
gives rise to a map
T (GV) ^ T (GW) ! T (GV W);
which defines a multiplication on MUG . The unit element is represented by the
maps SV ! T (GV) 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?.
Definition 3.3.Let V U be of dimension n. Then the classifying map V ! Gn
induces a map of Thom spaces SV ! T (Gn), which represents an element tV 2
MUGV -2nknown as a Thom class.
Proof of Proposition 3.2.We show that the Thom class tV is invertible. The iso-
morphism between MUGV(X) and MUG2|V(|X) is then given by multiplication by
this Thom class.
The class in MUG2n-Vrepresented by the map S2n ! T (GV) induced by the
classifying map Cn ! GVis the multiplicative inversenof tV . The product of thi*
*s_
class with tV is homotopic to the unit map SV C ! T (GV Cn). |__|
The most pleasant way to produce classes in MUG* is from equivariant stably
almost complex manifolds. Recall that there is an real analog of BUG (n), which
we call BOG (n), and which is the classifying space for all G-vector bundles.
6 DEV PRAKASH SINHA
Definition 3.4.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 equivariant bordism.
Definition 3.5.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 3.6.Define a map P T :U;G*! MUG*as follows. Choose a represen-
tative 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 bun*
*dle
with a tubular neighborhood of M in SV . Define P T ([M]) as the composite
SV !c T () T(f)!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 classifying map ! ||.
The proof of the following theorem translates almost word-for-word from Thom's
original proof.
Theorem 3.7. 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. A theorem of Comeza"na states that P T is split inj*
*ective
for abelian groups. The following classes illustrate the failure of the Pontrij*
*agin-
Thom map to be an isomorphism.
Definition 3.8.Compose the map SV ! T (Gn), in Definition 3.3 of the Thom
class 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*. The key difference between the equiva*
*ri-
ant and ordinary settings is the lack of transversality equivariantly. For exam*
*ple,
if V G = {0} the inclusion 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. Reca*
*ll
that ordinary homotopical bordism MU can be defined using Thom spaces as in
our definition of MUG but without any group action present.
Definition 3.9.Define the augmentation map ff: MUG ! MU by forgetting the
G-action on MUG . When G is abelian and H is a subgroup of G define resGHto
be the map from MUG*! MUH* by restricting the G-action to an H-action.
We need to have G abelian for the map resGHto be so defined. In the abelian
setting, any complex representation of H extends to a complex representation of
G, so that when its G-action is restricted to an H-action the Thom space T (Gn)
coincides with T (Hn).
COMPUTATIONS OF COMPLEX EQUIVARIANT BORDISM RINGS 7
Definition 3.10.Define the inclusion map : MU ! MUG by composing a map
Sn ! T (n) with the inclusion T (n) ! T (Gn).
On coefficients, defines an MU*-algebra structure on MUG*. The kernel of ff
on coefficients 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 the G-action. On the other hand, is injective, which follows f*
*rom
the following proposition which is proved for example in [14].
Proposition 3.11.The composite ff O : MU ! MU is homotopic to the identity
map.
4. The Connection Between Taking Fixed Sets and Localization
The connection between localization, in the commutative algebraic sense, and
"taking fixed sets" has been a fruitful theme in equivariant topology. We devel*
*op
this connection in the setting of bordism in this section.
The main goal of this section is to prove Theorem 1.1, which we restate here *
*for
convenience. Let R0 denote the sub-algebra of MUG*generated by the classes eV
and [P(n_ V )] as V ranges over non-trivial irreducible representations. Let S *
*be
the multiplicative set in R0 of non-trivial Euler classes.
Definition 4.1.A compact Lie group G is nice if every proper subgroup is con-
tained in a proper normal subgroup.
For example, abelian groups and p-groups are nice.
Theorem (Restatement of Theorem 1.1).Let G be a nice group. The inclusion of
R0 into MUG*becomes an isomorphism after inverting S.
We prove this theorem by first explicitly computing S-1MUG* and then com-
puting the images of generators of R0 in S-1MUG*. We start with the following
lemma, which provides translation between localization and topology. For any co*
*m-
mutative ring R and element e 2 R let R[1_e] denote the localization of R obtai*
*ned
by inverting e.
Lemma 4.2. As rings, ^MUG*(S1 V ) ~=MUG*[_1_eV]:
Proof.The left-hand side ^MUG*(S1 V ) is a ring because S1 V is an H-space via
the equivalence
S1 V ^ S1 V ~=S1 V :
To compute the left-hand side, apply ^MUG*to the identification S1 V = lim-!Sn*
*V .
After applying the suspension isomorphisms ^MUG*(SkV ) ~= M^UG*+|V(|Sk+1V ),
the maps in the resulting directed system are multiplication by the eV . *
* |___|
We will see that after inverting Euler classes, equivariant bordism is comput*
*able.
If G is a nice group and {Wi} are the non-trivial irreducible representations o*
*f G
then Z = Si(1 Wi) has fixed sets ZG = S0 while ZH is contractible for any
H G. Hence, our next lemma, taken with Lemma 4.2, establishes the strong link
between localization and taking fixed sets.
8 DEV PRAKASH SINHA
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 m*
*ap
Maps G(X; Y ^ Z) ! Maps(XG ; Y G)
is a homotopy equivalence.
Proof.The restriction map is a fibration whose fiber at a given point is the sp*
*ace
of G-maps which are specified on XG . Using the skeletal filtration of X, we can
then filter this mapping space by spaces
Maps G(Dk x G=H ; Y ^ Z);
such that the maps are specified on the boundary of Dk x G=H , and where H
is a proper subgroup of G. A standard change-of-groups argument yields that
this mapping space is homeomorphic to Maps(Dk; (Y ^ Z)H ), again with the map
specified on the boundary. But (Y ^ Z)H is contractible, and thus so are thes*
*e __
mapping spaces. Thus, the fiber of the restriction map is contractible. *
* |__|
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 choosing specific representatives of isomorphism classes of representations.*
* Let
Kn Kn+1 denote a sequence of representations which eventually contain all
irreducible representations infinitely often and such that Kn ? Kn+1 contains
precisely one copy of the trivial representation. If G is finite, we can let Kn*
* be the
direct sum of n copies of the regular representation.
Definition 4.4.Let X be a G-prespectrum. We define the geometric fixed sets
spectrum G X by passing from a prespectrum OEG X 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 m*
*aps
are composites
? G (K ? )G G G
(XKn )G -! (Kn XKn+1) -! n (XKn+1) = (XKn+1) ;
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.
While the prespectrum OEG (X) depends on the choice of filtration K*, the spe*
*c-
trum G X is independent of this choice.
Lemma 4.5. Let Z be as in Lemma 4.3. Then for any G-prespectrum X, the
prespectra (X ^ Z)G and G X are homotopy equivalent.
Proof.From the definition of (X ^ Z)G , consider
(W (XWV ^ Z))G :
Applying Lemma 4.3, the restriction from this mapping space to WG (XWV )G
is a homotopy equivalence. Choosing V = Kn, we see that WG (XWKn )G is an
entry of OEG X. The bonding maps clearly commute with these restriction to fixe*
*d_
sets maps, so we have an equivalence of spectra. |__|
Note that any Z as in Lemma 4.3 is an (equivariant) H-space as Z ^ Z ' Z.
Hence if X is a ring spectrum so is (X ^ Z)G . Taking Lemma 4.2 and Lemma 4.5
together, we have the following.
Proposition 4.6.Let G be a nice group and let S be the multiplicative set of no*
*n-
trivial Euler classes in MUG*. Then as rings S-1MUG*~=(G MUG )*.
COMPUTATIONS OF COMPLEX EQUIVARIANT BORDISM RINGS 9
To compute (G MUG )*, we can use the geometry of Thom spaces. Because
smashing a weak equivalence of prespectra with a complex yields another weak
equivalence, we have Z ^ MUG ' Z ^ T UG as prespectra, where T UG denotes the
equivariant Thom prespectrum and Z is as in Lemma 4.3. Hence,
G MUG ' Z ^ MUG ' Z ^ T UG ' G T UG ;
As required by the definition of G , we proceed with analysis of fixed-sets of *
*Thom
spaces.
We first need the following basic fact about equivariant vector bundles.
Proposition 4.7.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.
The following result is due to tom Dieck [7] in the case of the group G = Z=p.
Lemma 4.8. For any compact Lie group G, the G-fixed set of the Thom space of
Gnis homotopy equivalent to
0 1
_ Y
T (|WG |) ^ @ BU(V (W ))A ;
W2R+(G)n V 2Irr*(G) +
where we define R+ (G)n as the subset of dimension n representations in R+ (G)
and we recall that V (W ) is the greatest number m such that m V appears as a
summand of W .
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*
*.7
we see that this classifying space is weakly equivalent to
0 1
a Y
@ BU(V (W ))A :
W2R+(G) V 2Irr(G)
Over each component of this union, the universal bundle decomposesQas 1 x 2,
where 1 is the universal vector bundle over the factor of BU(n) corresponding
to the trivial representation. The fixed set 1 Gis all of 1 while the fixed set*
*_2_Gis
the zero section. The result now follows by passing to Thom spaces. |*
*__|
For convenience, we define the following spectrum.
Definition 4.9.Let _
S[Irr*(G)] = S2(|W|):
W2Z[Irr*(G)]R(G)
Define a ring spectrum structure on S[Irr*(G)] by sending the V summand smashed
with the W summand to the V + W summand.
Theorem 4.10. For any compact Lie group G,
Y
G MUG ' S[Irr*(G)] ^ MU ^ ( BU)+ :
V 2Irr*(G)
10 DEV PRAKASH SINHA
After Lemma 4.8, the proof of this theorem is straightforward,Qpassing from t*
*he
prespectrum OEG T UG to the spectrum S[Irr*(G)] ^ MU ^ ( V 2Irr*(G)BU)+ :
For a non-trivial irreducible representationQV , let fV be the map from CPk m*
*ap-
ping to the V th wedge summand of V 2Irr*(G)BU by the canonical inclusion to
BU(1) BU on the V th factor and by the trivialQmap on the other factors. Define
Yi;Vto be the class in the subgoup MU2(i-1)( V 2Irr*(G)BU)+ ) of (G MUG )2i
represented by CPi-1
We may now complete the central computation of this section.
Theorem 4.11. The ring (G MUG )* is a Laurent algebra tensored with a poly-
nomial algebra as follows:
-1
(G MUG )* ~=MU* eV ; eV ; Yi;V:
Here V ranges over irreducible representations of G, i ranges over the positive
integers, where as indicated by notation eV is the image of the Euler class eV 2
MUG*under the canonical map to the localization and where Yi;Vare as above.
Proof.This theorem is simple computation after Theorem 4.10. We use the compu-
tation MU*(BU) ~=MU*[Yi] as rings, where Yi is represented by CPi mapping to
BU via its inclusion into BU(1), which is standard as in [1]. BecauseQMU*(BU) is
a free MU*-module, it follows from the K"unneth theorem that MU*( Irr*(G)BU)
is a polynomial algebra as well. To finish the computation, we note that the Eu*
*ler
class eV maps to the class in of ss-|V |S[Irr*(G)] which is the generator_on th*
*e V th
summand. |__|
From Proposition 4.6 and the above theorem we have the following.
Corollary 4.12.S-1MUG*~=MU*[eV ; e-1V; Yi;V]:
We have shown the intimate relation between localization and taking fixed sets
for homotopical equivariant bordism. We will also need the following geometric
point of view, which dates back to Conner and Floyd.
Proposition 4.13.Let M be a tangentially complex G-manifold. The normal bun-
dle of 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.*
*7,
j|MG decomposes as a complex G-bundle
M
j|MG ~=j1 jae;
ae2Irr0(G)
where j1 has trivial G-action. But we can identify j1 as havingLunderlying real
bundle equal to T MG x Rk. So the normal bundle underlies ae2Irr0(G)jae,_
which gives the desired complex structure. |__|
As Comeza"na points out, this proposition would not be true if in the definti*
*on
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 cou*
*ld
only guarantee that normal bundles to fixed sets would be stably complex.
COMPUTATIONS OF COMPLEX EQUIVARIANT BORDISM RINGS 11
Definition 4.14.Let
0 1
M Y
F* = MU*-|W|@ BU(W (V ))A :
W2R+(G) V 2Irr*(G)
Define the homomorphism ': U;G*! F* as sending a class [M] 2 U;Gnto the
class represented by MG with reference map 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.
Proposition 4.15 (tom Dieck).The following diagram commutes
U;G* --'--! F*
?? ?
yPT ?yi
MUG* ----! (G MUG )*;
where i is the inclusion map
0 1 0 1
M Y M Y
MU*-|W|@ BU(W (V ))A ! MU*-|W|@ BUA :
W2R+(G) V 2Irr*(G) W2R(G) Irr*(G)
We may now compute the images of geometric classes in MUG*under localization
by geometric means.
Proposition 4.16.Let V be an irreducible representation of G. The image of
[P(n V )] in (G MUG )* is Yn;V+ X, where X is (eV *)-n for one-dimensional V
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 C*
*Pn-1,
whose normal bundle is the tautological line bundle over CPn-1Qtensored with V *
*. As
a class in (G MUG )*, this manifold with reference map to Irr*(G)BU represents
Yn;V. Alternately, when all other coordinates are zero the resulting submanifol*
*d is
the space of lines in V , which is an isolated fixed point when V is one-dimens*
*ional
and is a projective space with no fixed points, as V has no non-trivial invaria*
*nt
subspaces, when V has higher dimension. As classes in (G MUG )*, isolated fixed_
points represented negative powers of Euler classes. |_*
*_|
We finish this section by proving that for nice groups, the inclusion of R0 i*
*nto
MUG*becomes an isomorophism after inverting Euler classes.
Proof of Theorem 1.1.Recall that from Corollary 4.12 we have that for nice grou*
*ps
S-1MUG*~=MU*[eV ; e-1V; Yi;V]: It suffices to consider the image of S-1R0 in th*
*is
ring. The Euler classes and their inverses are in this image by definition. And*
* by
Proposition 4.16, the classes Yi;Vare in this image modulo negative powers_of E*
*uler
classes. |__|
12 DEV PRAKASH SINHA
5.Computation of MUG*
By Theorem 1.1, for nice groups G any x 2 MUG* can be multiplied by an
Euler class to get a class in R0 modulo the annihilator of some Euler class. Our
plan, which we carry out for abelian groups, is to build MUG*from R0 by division
by Euler classes. We are faced with two questions: "when can one divide by an
Euler class?" and "what are annhilators of Euler classes?" When G is a torus,
Theorem 1.2 answers both of these questions. Recall that K(V ) is the subgroup *
*of
T which acts trivially on V .
Theorem (Restatement of Theorem 1.2).The sequence
resTK(V )K(V )
0 ! MUT*.eV!MUT* ! MU* ! 0
is exact.
Proof.We construct the appropriate Gysin sequence and show that it breaks up
into short exact*sequences.
Apply ^MUT to the cofiber sequence S(V )+ !i S0 j!SV to get the long exact
sequence
2n j* i* ffi 2n+1
. .!.^MUT (SV ) ! MUT 2n! MU2nT(S(V )) ! ^MUT (SV ) ! . .:.
k
As MUT has suspension isomorphisms for any representation, ^MUT (SV ) ~=
MUk-VT. By Proposition 3.2, MUk-VT ~=MUk-2G. The map j* is by definition
multiplication by eV .
To compute MUkT(S(V )), we note that for a non-trivial irreducible representa*
*tion
of a torus S(V ) is homeomorphic to the orbit space T=K(V ). But maps from this
orbit space to MUT are in one-to-one correspondence with maps from a single poi*
*nt
in the orbit to the K(V )-fixed set of MUT, which is homeomorphic to the K(V )-
fixed set of MUK(V ). We deduce that MUkT(S(V )) ~=MUkK(V )and that i* is the
restriction map.
By Comeza"na's theorem (Theorem 1.3), both MU*Tand MU*K(V )are concen-
trated in even degrees. Hence the long exact sequence above yields the short_ex*
*act
of the theorem. |__|
Remark. Let T = S1 and ae be the standard representation, so the restriction to
the kernel of ae is the augmentation map. There is a geometric construction whi*
*ch
reflects the fact that, by Theorem 1.2, the augmentation ideal is principal, ge*
*nerated
by eae.
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 a null-homotopy. Construct
an S1-equivariant map f(F) :X x I x S1 ! Y by sending
(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.
COMPUTATIONS OF COMPLEX EQUIVARIANT BORDISM RINGS 13
As in the introduction, fix a splitting sV of resK(V )as a map of MU*-modules.
Let fiV = sV O resK(V.)And for any x 2 MUTnlet V (x) be the unique class in
MUTn+2such that eV . V (x) = x - fiV (x). The existence and uniqueness of V (x)
follow from Theorem 1.2 and the fact that x - fiV (x) is in the kernel of resK(*
*V ).
We are ready to prove our main theorem.
Theorem (Restatement of Theorem 1.5).For any choice of splittings sV , MUT*is
generated as an MU*-algebra over the operations V by the classes eV and [P(n_
V )], where n 2 N and in all instances V ranges over non-trivial irreducible co*
*mplex
representations of T . Relations are as follows:
1. eV V (x) = x - fiV (x)
2. V (fiV (x)) = 0
3. V (eV ) = 1
4. V (xy) = V (x)y + fiV (x)V (y) + V (fiV (x)fiV (y))
5. V W x = W V x + W V fiW (x) - W V (eW )fiV (W x)
- W V (fiV (eW )fiV (W x));
where V and W range over non-trivial irrreducible representations of T and x and
y are any classes in MUG*.
Proof.By Theorem 4.11, any class in MUT*can be multiplied by an Euler class to
give a class in R0 modulo the kernel of the canonical map from MUT*to S-1MUT*,
where S is the multiplicative set of non-trivial Euler classes. By Theorem 1.2,*
* the
kernel of the map from MUT*to the ring one obtains by inverting a single Euler
class is injective, so it follows that the map to S-1MUT*is injective. Hence, a*
*ny
class in MUT*is the quotient of some class in R0 by an Euler class.
Thus, we may filter MUT*exhaustively as
R* = R0* R1* . . .MUT*;
where Ri*is obtained from by adjoining to Ri-1*all x 2 MUT*such that x . eV =
y 2 Ri-1*for some V . By Theorem 1.2 the set of all such y for a given V is
Ker(resK(V )) \ Ri-1*. The kernel of the restriction map is clearly generated b*
*y all
classes y - fiV y. So we may obtain Ri*from Ri-1*by applying to every class in
Ri-1*, which proves that MUT*is generated over the operations V by R0.
Next, we note that the relations are readily verifyable. Relation 1 holds by
definition. And we may use the fact that multiplication by non-trivial Euler cl*
*asses
is a monomorphism to verify relations 2, 3 and 4 by multiplying them by eV , and
5 by multiplying it by eV eW .
We are left to show that the relations are complete. For convenience, if I =
V1; . .;.Vk is a k-tuple of representations let I(x) = VkVk-1. .V.1x. We fix an
ordering the representations of T .
We claim that a multiplicative generating set for MUT*is given by classes I(r)
where r is a generator of R0 and I is a (possibly empty) k-tuple ofQrepresentat*
*ions
which respects the ordering we have imposed, as well as classes I( fiVkxi), wh*
*ere
Vk is the minimimal representation in I. By the relation 4, to construct a gene*
*rating
set it suffices to consider classes V (x), where x is either primitive itself o*
*r a product
of classes in the image of fiV . And by relation 5, it suffices to consider wit*
*hin those
classes only the ones V (I(x)) where V is greater than any of the representatio*
*ns
in I.
Next we give an additive basis for MUT*. Fix an ordering on the generators of*
* R0
in which ri< rj if rj is in the image of some fiV where V is less than any W su*
*ch
14 DEV PRAKASH SINHA
that riis in the image of fiW . We define an additive basis for MUT*in twoQfami*
*lies.
Basis elements in the first family are the monomials I(r0)m, where m = ri
is a monomial in R0, I respects our ordering on the representations of T , r0 i*
*s a
generator of R0 which is not in the image of fiVk where Vk is the minimal eleme*
*nt
of I, under our ordering r0 is greater than any of the ri, and for each represe*
*ntation
V 2QI and each i we have that eV 6= ri. Basis elements in the second family are
I( ri), where I respects our ordering on the representations of T and each ri *
*is
a generator of R0 in the image of fiVk where Vk is the minimal element of I.
We check that this basis is linearly independent by mapping to S-1MUT*. Define
a multiplicative basis of S-1MUT*using the images of elements of R0 along with
the multiplicative inverses of Euler classes. Extend our ordering of generators*
* of
R0 to an ordering of generators of S-1MUT*in which the inverses of Euler classes
are less than any generator of R0. Now order the monomials in S-1MUT*by a
dictionary ordering. Then the image of an additive basis element as defined abo*
*ve
Y Y
ri. e-1Vi+ lower order terms:
Vi2I
These images are clearly linearly independent.
Finally, we show that the relations suffice to reduce any product of multipli*
*cative
generators to a linear combination of additive basis elements. Let m be a monom*
*ial
in the multiplicative generators defined above. If m = eV . I(x) . . .;.with V *
*2 I,
we may use relation 5 to express I(x) as a sum of terms V (J(xi)) and then use
relation 1 to simplify. We may thus reduce so that for each representation V we
do not have both eV and V appearing in m. Next, note that 4 gives rise to the
following relation
(1) xV (y) = V (x)(y - fiV (y)) + fiV (x)V (y):
We may use this relation repeatedly so that all of the operations which appear *
*in
m are appliedQto a single generator of our choosing. So we reduce to terms of t*
*he
form I(r0) ri, where r0 is greater than any ri in our ordering of generators *
*of
R0. Finally, we use the relation 5 to reorder theQrepresentations which appear *
*in
I. Note that when we do soQwe may get terms I( fiV (xi))y which violate one of
our conventions in that fiV (xi) could be less than some generators which app*
*ear
in y. We may then use equation 1 so that the operations are being applied to a
maximal generator, followed by relation 5 to reorder. This process terminates. *
*At
each stage we may associate a monomial in R0 to a product of our generators of
MUT*by forgetting all operations V . After an application of the relation 5 and
equation 1 this associated monomial will be strictly smaller for each term than*
* the
associated monomialQfor the original product. Once the associated monomials are*
* of
the minimal form fiVk(xi) where Vk is minimal among representations appearing
in the indexing set for operations, we may use 4 to equate the term with an_add*
*itive_
basis element in the second family. |__|
6.The Completion Map and a Construction of Conner and Floyd
From our computations, it is clear that the structure of MUG*is governed by t*
*he
operations V . We call these operations Conner-Floyd operations because in the
special case of G = S1, V = ae the standard representation, and [M] is a geomet*
*ric
class, there is a construction of ae([M]), which dates back to work of Conner a*
*nd
Floyd.
COMPUTATIONS OF COMPLEX EQUIVARIANT BORDISM RINGS 15
Definition 6.1.Define fl(M) for any stably complex S1-manifold to be the stably
complex manifold
___
fl(M) = M xS1 S3 t (-M ) x P(1 ae);
___
where S3 has the standard Hopf S1-action, -M is the S1-manifold obtained from
M by imposing a trivial action on M and taking the opposite orientation, and the
S1-action on M xS1 S3 is given by
(2) i . [m; z1; z2]= [i . m; z1;:iz2]
Inductively define fli(M) to be fl(fli-1(M)), where fl0(M) = M.
Proposition 6.2.Let ae be the standard representation of S1. And let M be a
stably complex S1-manifold. Then ae[M] = [fl(M)].
Proof.As1the localization map is injective it suffices to check_the_equality in
S-1MUS*. By Proposition 4.15 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 those of X
crossed with ae_(when in the notation of equation 2 above, m is fixed and z2 = *
*0)
along with an X crossed with ae (when z1 = 0). In the localization, crossing_wi*
*th_ae
coincides with multiplying by e-1ae. |__|
This geometric construction of a single Conner-Floyd operation allows us expl*
*icit
understanding of the most important representation of MUT*, namely the map from
MUT*to its completion at its augmentation ideal. As a special case of Theorem 1*
*.2,
we know that the augmentation ideal of MUS1*is principal, generated by eae. Be-
cause the augmentation map is split and multiplication by eaeis a monomorphism,
the completion of MUS1*at its augmentation ideal is a power series ring over MU*
where eaemaps to the power series variable under completion. As an immediate
consequence of Proposition 6.2 we have the following.
Theorem 6.3. Let [M] be class in MUS1*which is the image under the Pontrjagin-
Thom map of the class in geometric bordism1represented by the complex S1-manifo*
*ld
M. The image of [M] under the map from MUS* to its completion at its augmen-
tation ideal, which is isomorphic to MU*[[x]], is the power series
[ff(M)] + [ff(fl(M))]x + [ff(fl2(M))]x2 + . .;.
where ff(fli(M)) is the manifold obtained from fli(M) simply by forgetting the *
*G-
action.
Understanding this completion map for geometric classes is important for some
geometric applications. As mentioned in the introduction, Comeza/ na and May
have proved that for abelian G, (MUG*)^I(X) ~=MU*(X xG EG). So this comple-
tion homomorphism a the connection between MUG and any cohomology theory
which uses the Borel construction. For example, let ffl be a genus, that is a *
*ring
homomorphism from MU* to some ring E*. For an extensive introduction to gen-
era, see [15]. We may extend ffl to an equivariant genus U;G*! H*(BG) ^E*.
Given a G-manifold M, take M xG EG and use the genus ffl to produce a class
in H*(BG) E* by "integration over the fiber". In our setting, for G = S1, we
may define this equivariant genus by taking the image of a class [M] under com-
pletion, namely f 2 (MUS1*)^I~=MU*[[x]], and applying ffl term-wise to get a cl*
*ass
in H*(CP1 ) ^E* ~=E*[[x]].
16 DEV PRAKASH SINHA
A genus ffl is strongly multiplicative if for any fiber bundle of stably comp*
*lex
manifolds F ! E ! B, ffl(E) = ffl(F ).ffl(B). The following theorem is a fundam*
*ental
starting point in the study of genera, saying essentially that strongly mutlipl*
*icative
genera are rigid.
Theorem16.4. Let ffl be a strongly multiplicative genus. Then for any class [M*
*] 2
U;S*, the equivariant extension ffl([M]) is equal to ffl([ff(M)]) 2 E* E*[[x]].
Proof.By Theorem 6.3 the image of [M] under completion is
[ff(M)] + [ff(fl(M))]x + [ff(fl2(M))]x2 + . .:.
1
For any X 2 U;S*we have that ffl([ff(fl(X))] = 0 because ffl is strongly mulitp*
*licative
and by definition fl(X) is the difference between a twisted product and_a_trivi*
*al
product of X and CP1. |__|
Returning to computation of the completion map on MUT*, we now focus on
Euler classes.
Proposition 6.5.The image of the Euler class eaen in the completion (MUS1*)^I
is [n]F x, 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
equivariant cohomology theories, the Euler class of the bundle V over a point g*
*ets
mapped to the Euler class of V xG EG over BG . For G = S1, V = aen the resulti*
*ng
bundle is the nth-tensor power of the tautological bundle over BS1, whose Euler_
class is by definition the n-series. |__|
We are now ready to state our theorem about the image of the completion map
for MUT*. When T = (S1)k, the completion of MUT*at its augmentation ideal is
isomorphic to MU*[[x1; . .x.k]].
Definition 6.6.Let Yn(x) 2 MU*[[x]] be the image of the class [P(n_ ae)] under
the completion map.
Theorem 6.7. Let E be the set of all series
[m1]F x1 +F . .+.F[mk]F xk 2 MU*[[x1; . .x.k]]:
The image of MUT*in its completion at the augmentation ideal is contained in the
minimal sub-ring A of MU*[[x1; . .;.xk]] which satisfies the following two prop*
*er-
ties:
o E A, and A contains the series Yi(f) where f 2 E and Yi(f) are defined
above.
o If fff 2 S then ff 2 A, for any f 2 E.
We can recover the image of MUG*in its completion at the augmentation ideal
for general G by reducing MU*[[xi]] modulo the ideal ([di]F xi), where di are t*
*he
orders of the cyclic factors of G.
Proof.The first condition on A says that the image contains all images of class*
*es
in R0. Indeed, E is the image1of the Euler classes. And we check that the image
of [P(i aen )] in (MUS*)^Iis Yi([n]F x), which follows from the fact that the*
* S1
action on [P(iaen )] is pulled back from the S1 action on [P(iae)] by the degr*
*ee n
homomorphism from S1 to itself. By Theorem 4.11 we may build any class in MUT*
by dividing classes in R0 by Euler classes. The second condition on S accounts *
*for_
all possible quotients by Euler classes in the image. |*
*__|
COMPUTATIONS OF COMPLEX EQUIVARIANT BORDISM RINGS 17
Suppose f = a0+ a1x + a2x2+ . .i.s the image of x 2 MUS1*under completion.
Then the image of ae(x) under completion is that a1 + a2x + a3x2 + . .i.s in the
image. More generally, any ai+ ai+1x + . .i.s in the image of the completion ma*
*p.
So the property of a series being in the image of the completion map depends on*
*ly
on the tail of the series. It would be interesting to find an "analytic" way to*
* define
this image.
7.Applications and Further Remarks
In this section we give an assortment of applications and indicate directions*
* for
further inquiry.
Our first application is in answer to a question posed by Bott. Suppose a gro*
*up
acts on a manifold compatible with a stably complex structue and that the fixed
points of the action are isolated. What can one say about the representations
which appear as tangent spaces to the fixed points? If there are only two fixed
points, the representations must be dual, which one can prove by Atiyah-Bott
localization. What happens for three or more fixed points is an active area of *
*inquiry
in equivariant cohomology. With our bordism techniques, we can get answer some
of these questions, as wel as go beyond local information.
Theorem (Restatement of Theorem 1.6).Let M be a stably-complex four dimen-
sional S1-manifold with three isolated fixed points. Then M is equivariantly co*
*bor-
dant to P(1_ V W ) for some distinct non-trivial irreducible representations V
and W of S1.
Proof.For convenience, let refer to the Euler class eaen 2 MUS1*by en. A complex
S1 manifold M with three isolated fixed points defines a class in MUS1*whose
image under : MUS1*! S-1MUS1*is
([M]) = e-1ae-1b+ e-1ce-1d+ e-1fe-1g
for some integers a; . .;.g. We let T denote
1 -1 S1
ea . .e.g. [M] = ecedefeg + eaebefeg + eaebeced 2 MUS* S MU* :
Without loss of generality, assume a1is greatest of the integers a; . .;.g in*
* absolute
value. As T is divisible by ea in MUS*, Theorem 1.2 implies that T restricted to
MUZ=a*must 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 be equal to a. We first claim that*
* this
number must be -a.
Let S^adenote the multiplicative set generated by all the Euler classes assoc*
*iated
to irreducible representations except for ea. By localizing the modules in The*
*o-
rem 4.11 and Theorem 1.2, we find that S-1^aMUT*is generated over the operation
a by S-1^aR0. Suppose that |b|; |d|; |f|; |g| < |a| and that c = a. Then
e-1ae-1b+ e-1ae-1d+ e-1fe-1g
is in the image of the canonical map from S-1^aMUS1*to S-1^aR0, as it is actual*
*ly in
the image of . Then we must have that e-1b+ e-1dis divisible by ea and thus is
zero in S-1MUZ=a*where S here is the multiplicative set of all Euler classes of*
* Z=a.
This localization of MUZ=a*is the the target of the restriction map from S-1^aM*
*US1*.
And by abuse we are using the same names for Euler classes for different groups.
But because |b|; |d| < |a|, e-1b, e-1dand their sum are non-zero in S-1MUZ=a*.
18 DEV PRAKASH SINHA
It is straightforward to rule out cases where some of |b|; |d|; |f|; |g| are *
*equal to
|a|.
Next, consider the class
1
^a([M]) - ^a([P(1 + aea )])e-1d2 S-1^aMUS*;
where ^ais the canonical map to this localization. Its image under the map to t*
*he
full localization is
e-1ae-1b- e-1ae-1d+ e-1fe-1g;
which implies that e-1b- e-1dis divisible by ea in S-1^aMUS1*or that b d (mod *
* a).
But because |b|; |d| < |a| we have that d = a b depending on whether b is posi*
*tive
or negative.
Finally, as c = -a and d = b - a consider ([M] - [P(1 aea aeb )]), which
will be equal e-1fe-1g- e-1a-be-1-b. Case analysis of necessary divisibilities *
*as we have
been doing implies that this difference must be zero, so that the fixed-set dat*
*a of
[M] is isomorphic to that of P(1 aea aeb ).
Finally, by because the localization map is injective, this fixed-set data d*
*eter-
mines [M] as in S1-equivariant homotopical bordism uniquely, so that [M] must
equal [P(1 aea aeb )] in MUS14. But a theorem of Comeza"na says that the
Pontrijagin-Thom map from U;A*to MUA*is injective for abelian groups A. Hence
M is cobordant to P(1 aea aeb ). __
|__|
Our next 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 prime 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]F x modulo [n]F x in MU*[[x]]. Define ai2 MU* by (Q(x))k = a0+a1x+a2x2+
. ...Then
[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 MrMG has
a free G-action then [@(MG )] = 0 in MU*(BG ), where @(MG ) is the boundary
of a tubular 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 re*
*lation
among spheres with free G-actions.
Let ff0 = qk where q is a quotient of eaeby eaem in MUZ=n*. Inductively, let *
*ffi
be a quotient of ffi-1- ____ffi-1by eae(note that this quotient is not unique a*
*s we are
working in Z=n equivariant bordism. 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 t*
*hat
[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]F x is (Q(x))k
from which we can read off that __ffi= ai.
COMPUTATIONS OF COMPLEX EQUIVARIANT BORDISM RINGS 19
Note that our expressions in MU*(BZ=n) are independent of the indeterminacy_
in 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 stable homotopy theory [8]. Moreover, by work of Greenlees and May,
the theories we have been studying provide a unified framework in which to study
the characteristic classes E*(BG) for any complex-oriented theory E. We hope
that our understanding of relevant commutative algebra can lead to new insights
into these techniques.
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Department of Mathematics, Brown University, Providence, RI 02906
E-mail address: dps@math.brown.edu