REAL EQUIVARIANT BORDISM AND STABLE
TRANSVERSALITY OBSTRUCTIONS FOR Z=2
DEV SINHA
In this paper we compute homotopical equivariant bordism for the group Z=2,
namely MOZ=2*, geometric equivariant bordism NZ=2*, and their quotient as NZ=2*-
modules. This quotient is a module of stable transversality obstructions, clos*
*ely
related to those of [4]. In doing these computations, we use the techniques of *
*[7].
Because we are working in the real setting only with Z=2, these techniques simp*
*lify
greatly. Note that NZ=2*has been computed in [1] but the computation here of
MOZ=2*is new and in fact simpler than that of NZ=2*, as with a modern viewpoint
we make strong use of localization.
The paper is structured as indicated by section titles; we give basic definit*
*ions,
then statements of theorems, and finally proofs. Note that there is an unevenne*
*ss
in the level of exposition between these sections. The statements of theorems a*
*re
aimed more at experts in a first reading. The proofs are aimed at novices.
Thanks go to Peter Landweber for a close reading of this paper.
1.Definitions
Though all of the basic constructions can be made for an arbitrary compact Lie
group, we state things only for the group Z=2. For a more thorough introduction*
* see
chapter fifteen in [6] by Costenoble. Let o denote the trivial one-dimensional *
*real
representation of Z=2, and let oe denote the non-trivial one in which the non-t*
*rivial
element acts on R by multiplication by -1. Let BOZ=2(n) be the Grassmannian
of n-dimensional subspaces of U = 1 (o oe), with Z=2 action inherited from
U. And let Z=2nbe the universal Z=2 n-dimensional bundle over BOZ=2(n), and
T (Z=2n) be its associated Thom space. Given a representation V , let SV be t*
*he
one-point compactification of V .
Let NZ=2*denote the ring of bordism classes of Z=2-manifolds (manifolds with *
*an
involution), where bordism is defined in the usual way using manifolds with bou*
*nd-
ary as in [3]. To define homotopical bordism, first note that for any represent*
*ation
V there are maps SV ^ T (Z=2n! T (Z=2n+|Vd|efined by passage to Thom spaces of
the map classifying V x Z=2n. For non-negative n, MOZ=2nis the colimit
h Z=2 i
lim-!WSnoW ; T (Z=2|W|);
where [; ]Z=2 denotes the set of Z=2-maps, which in this case of taking maps fr*
*om
spheres is an abelian group. As in the ordinary setting, the Pontryagin-Thom
construction gives rise to a map NZ=2n! MOZ=2n(again see [3]). But we will see
that this map is not an isomorphism. Indeed, we extend the definition of MOZ=2n
1
2 DEV SINHA
to negative degrees in the standard way as
h Z=2 i
lim-!WSW ; T (Z=2|W|+|n|):
We will see that these groups are non-zero for any n, whereas NZ=2nare zero
for negative n by definition. We may in fact define an equivariant spectrum
MOZ=2, equipped with deloopings for any formal difference of representations of
Z=2, giving rise to associated homology and cohomology theories where MOZ=2*~=
MOZ=2*(pt:) ~=MO-*Z=2(pt:). There are periodicity isomorphisms (see [6]) which *
*im-
ply that MOZ=2V -Wdepends only on the virtual dimension of V - W , so we restri*
*ct
our attention to integer gradings.
The difference between geometric and homotopical bordism arises from the
breakdown of transversality in the presence of a group action. The most importa*
*nt
examples of classes in MOZ=2*not coming from geometric bordism are the Euler
classes (indeed, for Z=2 we will see that these are essentially the only exampl*
*es). The
representation oe defines a Z=2 vector bundle over a point by projection. There*
* are
no non-zero equivariant sections of this bundle. The Euler class eoe2 MO1Z=2(pt*
*:)
reflects the equivariant non-triviality of this bundle. Explicitly, given a rep*
*resenta-
tion V define the Euler class eV to be class of the composite S0 SV ! T (Z=2|V*
*)|in
MOZ=2-|V,|where the second map is defined by passing to Thom spaces the inclusi*
*on
of V as a fiber of |V |.
2. Statements of Theorems
Euler classes play an important role at every step in this paper. Tom Dieck [*
*5],
refining ideas of Atiyah and Segal, showed that localization by inverting these*
* Euler
classes corresponds to "reduction to fixed sets". We take tom Dieck's work as a
starting point, translating his results from the complex setting. Once we are t*
*aking
geometric fixed sets, we can make explicit computations. It is through explicit
computations that we prove the following key result.
Let P(V ) be the projective space of one-dimensional subspaces of a represent*
*ation
V , with inherited Z=2-action. And let [P(V )] 2 MOZ=2|Vb|-1e the image of the
bordism class of P(V ) under the Pontryagin-Thom map. Let R* be the sub-algebra
of MOZ=2*generated by the Euler class eoeand [P(no oe)], as n ranges over natu*
*ral
numbers. And let S be the multiplicative set in R* generated by eoe. By abuse, *
*use
S to denote the same multiplicative set in MOZ=2*.
Theorem 2.1. The canonical map S-1R* ! S-1MOZ=2*is an isomorphism.
In other words, any class in MOZ=2*can be multiplied by some power of eoeto
get a class in R* modulo the kernel of the localization map (which is in fact z*
*ero by
the next theorem). Hence, to understand MOZ=2*it suffices to understand R* and
division by eoe. Having computed S-1MOZ=2*we can also deduce from the proof of
Theorem 2.1 that R* is a polynomial algebra. In fact, R* is a maximal polynomial
subalgebra of MOZ=2*.
Remarkably, divisibility by eoeis closely tied to the interplay between equiv*
*ariant
and ordinary bordism. Let ff: MOZ=2*! N* be the augmentation map, that is the
REAL EQUIVARIANT BORDISM AND STABLE TRANSVERSALITY OBSTRUCTIONS FOR Z=23
map which forgets Z=2-action. Clearly eoeis in the kernel of ff. In some sense,*
* this
Euler class accounts for the entire kernel.
Theorem 2.2. The sequence
0 ! MOZ=2*.eoe!MOZ=2*ff!N* ! 0
is exact.
Using the exactness of this sequence, we define an operation on MOZ=2*. First
note that the augmentation map has a splitting : N* ! MOZ=2*which we may
define by taking a geometric representative for a class, imposing a trivial act*
*ion,
and passing to homotopical bordism through the Pontryagin-Thom map.
Definition 2.3.For any x 2 MOZ=2*define __xto be O ff(x). Then x - __xis in the
kernel of ff, so we define (x) to be the unique class such that eoe(x) = x - __*
*x.
Directly from Theorem 2.1 and Theorem 2.2 we may deduce the following.
Theorem 2.4. MOZ=2*is generated over N* by eoeand classes i([P(no oe)])
where i and n range over natural numbers. Relations are
o eoe(x) = x - __x
o (xy) = (x)y - __x(y);
where x and y range over the classes i([P(no oe)])
Equivariant bordism rings have been notoriously difficult to describe. We see*
* here
that MOZ=2*does not exhibit familiar properties. For example, any multiplicative
generating set of MOZ=2*must have a proper subset which is also a multiplicative
generating set.
When x is a geometric class, that is x is in the image of some [M] under the
Pontryagin-Thom map, there is a geometric construction of (x), a construction
which dates back to work Conner and Floyd [3].
Definition 2.5.Let fl(M) = M xZ=2S1, where S1 = {(x; y) 2 R2|x2 + y2 = 1}
has antipodal Z=2 action and Z=2 acts on the quotient by the rule g . [m; x; y]*
* =
[m; -x; y], where g is the non-trivial element of Z=2 and the brackets denote t*
*aking
equivalence classes.
Theorem 2.6. ([M]) = [fl(M)].
Because we have a geometric model for this operation, and because we may for
Z=2 identify geometric equivariant bordism as a sub-ring of homotopical equivar*
*iant
bordism, we also have explicit understanding of the geometric theory.
Theorem 2.7. The geometric Z=2 bordism ring NZ=2*is the sub-ring of MOZ=2*
generated over N* by the classes [fli(P(no oe))])
Finally, we may identify the quotient MOZ=2*=NZ=2*, which represents some sta*
*ble
transversality obstructions, as a NZ=2*-module.
Theorem 2.8. The NZ=2*-module MOZ=2*=NZ=2*is generated_by classes xk, k 2 N,
in degree -k, with relations [fl(M)]xk = [M]xk-1 - [M ]xk-1. The class xk is the
image under the canonical map to the quotient of the Euler class ekoe.
4 DEV SINHA
Hence the Euler classes are essentially the only transversality obstructions *
*in
*
* 2
this setting. This is not true in general, as for Z=3 for example the class Sae*
*z7!z!
Sae2 ! T (Z=32) is a non-trivial class in the quotient of homotopical and geome*
*tric
bordism which is not a multiple of an Euler class.
3.Proofs
As should be expected, our computations start with exact sequences due to
Conner and Floyd, and tom Dieck.
3.1. The Conner-Floyd Exact Sequence. Our goal is to prove the following
theorem, and then explicitly compute some of terms.
Theorem 3.1 (Conner-Floyd).There are maps j, OE and ffi, to be defined below, so
that the sequence
. .!.N*(BZ=2) j!NZ=2*OE!kN*-k(BO(k)) ffi!N*-1(BZ=2) ! . . .
is exact.
Historically, Conner and Floyd developed their exact sequence geometrically. *
*We
proceed in what is in some sense reverse of historical order, starting with the*
* long
exact sequence in NZ=2*of the based pair (c(EZ=2 )+ ; EZ=2 +), where EZ=2 is a
contractible space on which Z=2 acts freely (for example, the unit sphere in 1 *
*oe)
and c(EZ=2 )+ denotes the cone on EZ=2 with a disjoint basepoint added. To be
complete, we include the definition of relative bordism.
Definition 3.2.Let (X; A) be an admissible pair of Z=2-spaces. A singular Z=2-
manifold with reference to (X; A) is a pair (M; f) of a Z=2-manifold with bound*
*ary
M and a map f :M ! X such that f(@M) A. Two singular Z=2-manifolds
(M1; f1) and (M2; f2) are bordant when there is a singular Z=2-manifold (W; g)
such that M1 t M2 is Z=2-diffeomorphic to a codimension zero sub-manifold of
@W , g|M1 = f1, g|M2 = f2, and g(@W - (M1 t M2)) A.
Definition 3.3.Let NZ=2n(X; A) denote the group of equivalence classes up to bo*
*r-
dism of singular Z=2-manifolds with reference to (X; A).
After elementary identifications, the exact sequence associated to the pair (*
*c(EZ=2 )+ ; EZ=2 +)
reads
(1)
. .!.]NZ=2*(EZ=2 +) i*!NZ=2*j*!NZ=2*(c(EZ=2 ); EZ=2 ) @!N]Z=2*-1(EZ=2 +) ! . .:.
We start with analysis of the term NZ=2*(c(EZ=2 ); EZ=2 ) and the map j*. Form
the bordism module of Z=2-manifolds with free boundary, where we define the
bordism relation as in Definition 3.2.
Proposition 3.4.The module NZ=2*(c(EZ=2 ); EZ=2 ) is naturally isomorphic to
the bordism module of Z=2-manifolds with free boundary.
Proof.Any singular manifold with reference to (c(EZ=2 ); EZ=2 ) must have free
boundary. Conversely, given a Z=2-manifold with free boundary, there is no ob-
struction to constructing a reference map to (c(EZ=2 ); EZ=2 ). Applying these
observations to manifolds which play the role of bordisms, we see that these co*
*rre-_
spondences are well-defined up to bordism and inverse to each other. |*
*__|
REAL EQUIVARIANT BORDISM AND STABLE TRANSVERSALITY OBSTRUCTIONS FOR Z=25
Up to bordism, a Z=2-manifold with free boundary depends only on its fixed-set
data.
Proposition 3.5.A Z=2-manifold with free boundary M is bordant to any smooth
neighborhood N (MZ=2) of the fixed set of M as Z=2-manifolds with free boundary.
Proof.Let W = M x [0; 1], with "straightened angles". Then @W is free outside_of
M x 0 and N (MZ=2) x 1, so W is the required bordism. |__|
Thus the map j*: NZ=2*! NZ=2*(c(EZ=2 ); EZ=2 ) "reduces to fixed sets" in the
sense of sending a representative of a bordism class M to the bordism class of
smooth neighborhoods of its fixed set. If we choose N (MZ=2) to be a tubular
neighborhood of MZ=2 then we can use standard equivariant differential topology
to identify this tubular neighborhood with a Z=2-vector bundle over the fixed s*
*et,
where the action of Z=2 is free away from zero.
Proposition 3.6.NZ=2*(c(EZ=2 ); EZ=2 ) ~=kN*-k(BO(k)).
Proof.Use the identifications we have made so far to equate NZ=2*(c(EZ=2 ); EZ=*
*2 )
with the bordism module of Z=2-vector bundles over trivial Z=2-spaces where the
action is free away from zero. The fiber of a Z=2-vector bundle over a trivial *
*Z=2-
space is a representation. Because Z=2 has only one non-trivial representation,
the action on any fiber and thus the total space is completely determined. Hence
the forgetful map from this bordism module of Z=2-vector bundles which are free
away from the zero section to the the bordism module of vector bundles is an
isomorphism. The result follows from the fact that BO(k) is the classifying spa*
*ce
for vector bundles. Note that we must grade according to the dimension of the_
total space of the bundle in question. |__|
Interpreting the term ]NZ=2*(EZ=2 +) is more immediate. From the observation
that any singular manifold mapping to EZ=2 must itself have a free Z=2-action, *
*we
see that this module is isomorphic to the bordism module of Z=2-manifolds with
free Z=2-action. The bordism module of Z=2-manifolds with free Z=2-action has a
non-equivariant interpretation.
Proposition 3.7.The bordism module of Z=2-manifolds with free Z=2-action is
isomorphic to N*(BZ=2 ).
Proof.Consider the following diagram:
fM --f"--!EZ=2
?? ?
y ?y
M --f--! BZ=2 :
Given a representative M with reference map f to BZ=2 , pull back the principal
Z=2-bundle EZ=2 to get fM, which is in fact a free Z=2-manifold. Conversely,
starting with a free Z=2-manifold fM, there is no obstruction to constructing a*
* map
"fto EZ=2 . Pass to quotients to obtain f :M ! BZ=2 .
These maps are well-defined, as we apply the previous argument to the manifol*
*ds__
which act as bordisms. The composites of these maps are clearly identity maps. *
* |__|
Corollary 3.8.The module ]NZ=2*(EZ=2 +) is isomorphic to N*(BZ=2 ).
6 DEV SINHA
We may new deduce Theorem 3.1. We now give more geometric definitions of
the maps in this exact sequence.
Definition 3.9.Let j :N*(BZ=2 ) ! NZ=2*denote the N*-module homomorphism
which, using the identification of Proposition 3.7, sends a free Z=2-bordism cl*
*ass to
the corresponding Z=2-bordism class.
Proposition 3.10.The homomorphism j coincides with the homomorphism i* of
the exact sequence 1 under the isomorphism of Corollary 3.8.
Definition 3.11.Let ': NZ=2*! kN*-k(BO(k)) be the map which sends a class
[M] to the fixed set of [M] with reference map to tBO(k) classifying the normal
bundle to the fixed set.
Proposition 3.12.The homomorphism ' coincides with the homomorphism j* of
the exact sequence 1 under the isomorphism of Proposition 3.6.
Finally, we identify the boundary map.
Definition 3.13.Let ffi :k N*-k(BO(k)) ! N*-1(BZ=2 ) be the map of N*-
modules which sends E, a vector bundle, to the sphere bundle of E with fiberwise
Z=2 action defined by letting the non-trivial element of Z=2 act by multiplicat*
*ion by
-1.
Proposition 3.14.The homomorphism ffi coincides with the homomorphism @ of
the exact sequence of Equation 1 under the isomorphisms of Proposition 3.6 and
Corollary 3.8.
The proofs of Propositions 3.10, 3.12 and 3.14 are straightforward.
By Thom's seminal work, we can identify N*(BZ=2) and kN*-k(BO(k)) given
the standard computations of the mod 2 homology of BZ=2 and BO(k).
Proposition 3.15.N*(BZ=2) is a free N*-module generated by classes xi in de-
gree i, where i ranges over natural numbers. As the bordism module of free Z=2-
manifolds, the generator xiis represented by the i-sphere with antipodal Z=2-ac*
*tion.
Proof.The mod 2 homology of BZ=2 = RP1 is well known to be Z=2 in every
positive dimension. The class in dimension i is the image of the fundamental cl*
*ass
of RPi under inclusion. Under the identifications of Proposition 3.7 these clas*
*ses_
correspond to spheres with antipodal action. |__|
Next note that there are classifying maps for direct sums of the associated u*
*ni-
versal bundles BO(k) x BO(l) ! BO(k + l). These maps give rise to an H-space
structure on tBO(k), which in turn gives rise to a multiplication on homology.
Proposition 3.16.As a ring, kN*-k(BO(k)) is a polynomial algebra over N*
on classes bi2 Ni-1(BO(1)) represented by the tautological line bundle over RPi*
*-1.
The fact that this ring is a polynomial algebra follows from the mod 2 homo*
*l-
ogy computation, which is standard. The fact that generators are represented by
projective spaces is a straightforward Stiefel-Whitney number computation.
3.2. The tom Dieck Localization Sequence. Tom Dieck realized that there
was a connection between the Conner-Floyd exact sequence and the localization
methods in equivariant K-theory of Atiyah and Segal. This connection has been
fundamentally important in our work.
REAL EQUIVARIANT BORDISM AND STABLE TRANSVERSALITY OBSTRUCTIONS FOR Z=27
The following lemma provides translation between localization and topology.
Once again, let S be the multiplicative subset of MOZ=2*generated by eoe.
Lemma 3.17. As rings, M^OZ=2*(S1 oe) ~=S-1MOZ=2*:
Proof.Apply M^OZ=2*to the identification S1 oe = lim-!Snoe . After applying the
suspension isomorphisms M^OZ=2*(Skoe) ~=M^OZ=2*+1(Sk+1oe), the maps in the re- *
*__
sulting directed system are multiplication by the eoe. |*
*__|
Consider the cofiber sequence S(1oe)+ ! S0 ! S1oe, which is a model for
the sequence EZ=2 + ! S0 ! ^EZ=2, essentially our sequence of a pair from the
previous section. By the previous lemma, after applying M^OZ=2* to this cofiber
sequence, the second map in this sequence is the canonical map from MOZ=2*to
S-1MOZ=2*. We now identify the outside terms in this sequence.
Theorem 3.18. M^OZ=2*(^EZ=2) ~=k2ZN*-k(BO).
Proof.Recall the definition of MOZ=2*(^EZ=2) and consider the space of maps from
SV to ^EZ=2^ T (Z=2n), for any representation V .
First we show that for any Z=2-spaces X and Y , the restriction map
Maps Z=2(X; ^EZ=2^ Y ) ! Maps(XZ=2; (^EZ=2^ Y )Z=2) = Maps(XZ=2; Y Z=2)
is an equivalence. First note that this restriction is a fibration. Over a gi*
*ven
component of Maps(XZ=2; Y Z=2) a fiber is going to be the space of maps from X *
*to
^EZ=2^Y which are specified on XZ=2. We filter this mapping space by filtering *
*X.
Because the maps are already specified on XZ=2, we need only adjoin cells of the
form G x Dn, where G denotes Z=2 acting on itself by left multiplication. Hence
the subquotients in this filtration will be spaces of equivariant maps from Z=2*
*xDn
to ^EZ=2^ Y whose restriction to the boundary of Z=2 x Dn is specified. Because
Z=2 x Dn is a free Z=2-space, it suffices to consider the restriction of such a*
* map
to one copy of Dn. But ^EZ=2^ Y is contractible, hence so is this mapping space.
Therefore the fibers of our restriction map are contractible.
Applying this argument for X = SV , Y = T (Z=2n) we see that ourWcomputation
follows from knowledge of T (Z=2n)Z=2. We claim that T (Z=2n)Z=2= T (i)^BO(n-
i). We show this by analysis of the fixed set of Z=2n. Any fixed point of Z=2nm*
*ust lie
over a fixed point of BOZ=2(n). But the fixed set of BOZ=2(n) is the classifyin*
*g space
for Z=2-vector bundles over trivial Z=2-spaces. A vector bundle over a trivial *
*Z=2-
space decomposes as a direct sum according to decomposition of fibers according
to representation type. As there are only two representations types for Z=2, we
deduce that (BOZ=2(n))Z=2 = tBO(i) x BO(n - i). Restricted to a component
of this fixed set Z=2nwill be (i) x (n - i) where Z=2 fixes all points in the f*
*irst
factor and acts by multiplication by -1 on fibers in the second factor. Hence o*
*ne
component of the fixed set of Z=2nwill be (i) x BO(n - i).
Passing to Thom spaces we find
_
T (Z=2n)Z=2= T ((i)) ^ BO(n - i)+
8 DEV SINHA
. Using this result along with our first reduction we see that
Z=2 _
[SV ; S1oe^ T (Z=2n]Z=2= [SV ; T ((i)) ^ BO(n - i)+ ]:
The theorem follows by passing to direct limits. |__*
*_|
As M^OZ=2*(^EZ=2) ~=S-1MOZ=2*we are interested in multiplicative structure
as well.
Corollary 3.19.S-1MOZ=2*~=N*[xi; e; e-1], where as elements of kN*-k(BO),
xi are the images of the generators given in Proposition 3.16 under the canonic*
*al
inclusions of BO(k) into BO, and as an element of S-1MOZ=2*, e is the image of
the Euler class eoe.
Proof.The multiplication on MOZ=2*is defined by the maps T (Z=2i) ^ T (Z=2j) !
T (Z=2i+j) which are the passage to Thom spaces of the map classifying the prod*
*uct
W
of universal bundles. These maps restrict to T (Z=2j)Z=2= T ((i)) ^ BO(n - i)+
as the standard multiplication on T ((i)) factors smashed with the classifying *
*map
for Whitney sum on BO(n - i)+ factors.
Passing to the direct limit and neglecting grading, we are computing N*(ZxBO),
where ZxBO has an H-space structure which is the product of the group structure
on Z and the H-space structure on BO arising from Whitney sum. The computation
follows from the K"unneth theorem, as N*(Z) is a Laurent polynomial ring, which*
* by
our grading conventions is generated by a class we call e-1 in degree 1, and N**
*(BO)
is a polynomial ring in classes xiwhere xiis the image of the generator of Ni(B*
*O(1))
under the inclusion from BO(1) to BO. That e is the image of eoefollows directl*
*y __
from their definitions, chasing through the identifications of Theorem 3.18. *
* |__|
Next we identify the term which gives the kernel and cokernel of the localiza*
*tion
map.
Theorem 3.20. fM*(EZ=2 +) ~=N*(BZ=2 ).
Proof.This theorem is immediate as an application of Adams' transfer. But in the
spirit of giving elementary proofs, we argue geometrically as follows.
From the definition of fM*(EZ=2 +) consider a Z=2-map from SV to EZ=2 +
^T (Z=2n). Because the latter space has a free Z=2-action away from the basepoi*
*nt,
(SV )Z=2 must map to the basepoint. If we pass to the map from the quotient
SV =(SV )Z=2, we have a Z=2-map between Z=2-spaces which are free Z=2-manifolds
away from their basepoints. Because transversality is a local condition, it is *
*easy to
verify that transversality arguments hold in the presence of free Z=2-actions. *
*Given
a Z=2-map from SV =(SV )Z=2 to EZ=2 + ^T (Z=2n) we may homotop it locally to
a map which is transverse regular to the zero section of T (Z=2n) and pull back*
* a
sub-manifold of SV =(SV )Z=2which must necessarily be free.
So following classic techniques we identify M^OZ=2*(EZ=2 +) with the bordism__
module of free Z=2-manifolds. The theorem follows from Proposition 3.7. |_*
*_|
As one should suspect at this point, the tom Dieck localization sequence is p*
*re-
cisely the homotopical version of the Conner-Floyd sequence. The following theo-
rem is due to tom Dieck in the complex setting.
REAL EQUIVARIANT BORDISM AND STABLE TRANSVERSALITY OBSTRUCTIONS FOR Z=29
Theorem 3.21. The diagram
. .-.---! N*(BZ=2 )----! NZ=2* ---'-! k2NN*-k(BO(k)) ---ffi-!. . .
?? ? ? ?
y id?y PT ?y ?y
-1 @
. .-.---! N*(BZ=2 )----! MOZ=2* --S--! k2ZN*-k(BO) ----! . .;.
where the first vertical map is the identity map, the second is the Pontryagin-*
*Thom
map and the third is defined by the standard inclusion of BO(k) into BO, commut*
*es.
Proof.The proof of this proposition is almost immediate, as Pontryagin-Thom map
P T is a natural transformation of equivariant homology theories and our exact
sequence of a pair from which we defined the Conner-Floyd sequence coincides
with the cofiber sequence from which we defined the tom Dieck sequence. That
the Pontryagin-Thom map is an isomorphism when smashed with EZ=2 + follows
from the fact that transversality arguments carry through in the presence of a *
*free
Z=2-action. That the third vertical map is the standard inclusion follows from_*
*close
analysis of the Pontryagin-Thom map in this setting. |__|
3.3. Proofs of The Main Results.
Proof of Theorem 2.1.We show that the images of eoeand [P(no oe)] along with
e-1oegenerate S-1MOZ=2*. We do so using the explicit description of S-1MOZ=2*
from Corollary 3.19.
Consider the following diagram:
NZ=2* --'--! k2NN*-k(BO(k))
? ?
PT?y ?y
-1
MOZ=2* -S---! k2ZN*-k(BO) ~=N*[xi; e; e-1];
which combines results of Theorem 3.21 and Corollary 3.19. To compute the image
of [P(no oe)] under localization it suffices to look at fixed-set data, becaus*
*e it is a
geometric class.
There are two components of the fixed set [P(no oe)]Z=2. Using homogeneous
coordinates [y0; : :;:yn], these components are defined by the conditions yn = 0
and z1 = . .=.yn-1 = 0. The condition yn = 0 defines an n - 1 dimensional
projective space. Its normal bundle is the tautological line bundle. The condit*
*ion
y1 = . .=.yn-1 = 0 defines an isolated fixed point which has an n-dimensional
normal bundle. Using the generators named in Corollary 3.19 we have
S-1[P(no oe)] = xn + e-(n+1):
In Corollary 3.19 we also noted that the image of eoeunder localization was e.
It thus follows that the images of eoeand [P(no oe)] under localization, along*
* with
e-1oe, generate S-1MOZ=2*~=N*[xi; e; e-1], which is what was to be shown. *
*|___|
Proof of*Theorem 2.2.The exact sequence in question is a Gysin sequence. Apply
M^OZ=2 to the cofiber sequence G+ !i S0 j!Soe; where the first map is projection
of G onto the non-basepoint of S0. The resulting long exact sequence is
n j* i* * ffi n+1
. .!.M^OZ=2 (Soe) ! MOZ=2 n ! M^OZ=2 (G+ ) ! M^OZ=2 (Soe) ! . .:.
10 DEV SINHA
n
By the periodicity of MOZ=2, M^OZ=2 (Soe) ~= MOZ=2 n-2. By definition j* is
multiplication by eoe. From the fact that*Maps Z=2[G+ ; Y ] is homeomorphic to*
* Y
for any Z=2-space Y , we see that M^OZ=2 (G+ ) ~=N* and i* is the augmentation
map ff.
As we remarked after the statement of Theorem 2.2, the augmentation map ff
is split. Hence our long exact sequence breaks up into short exact sequences,_a*
*nd
the result follows. |__|
Proof of Theorem 2.4.First we verify relations. Then we show that the classes
listed generate MOZ=2*. Finally we show that the relations are a complete set o*
*f re-
lations. It is convenient to view MOZ=2*as a subring of S-1MOZ=2*~=N*[xi; e; e-*
*1],
which we can do as the previous theorem implies that eoeis not a zero divisor. *
*In
this way, we may verify the second family of relations by direct computation. T*
*he
first family of relations holds by definition.
For convenience, rename [P(no oe)] as Xn. By Theorem 2.1, any class in MOZ=2*
when multiplied by some power of eoeis equal to a class in R* modulo the annihi*
*lator
ideal of eoe, which is zero. Hence we may filter MOZ=2*exhaustively as
R* = R0* R1* . . .MOZ=2*;
where Ri*is obtained from by adjoining to Ri-1*all x 2 MOZ=2*such that x . eoe=
y 2 Ri-1*. By Theorem 2.2 the set of all such y is Ker(ff) \ Ri-1*. The kernel *
*of
the augmentation map is clearly generated by all classes y - __y. So we may obt*
*ain
Ri*from Ri-1*by applying to every class in Ri-1*. Since (xy) = (x)y - __x(y);
it suffices to apply only to primitive elements. It follows that eoeand i(Xn)
constitute multiplicative generators.
Finally, to show these relations are complete we identify an additive basis of
MOZ=2*. There are two types of monomials in the additive basis, those of the fo*
*rm
ekoef, f 2 N*[Xi] and those of the form k(Xj)f, f 2 N*[Xi|i > j]. We may check
that these classes are additively independent by mapping to S-1MOZ=2*. Define
the complication of a monomial in our basis elements to be the sum of the number
of times both eoeand appear in the monomial and the sum of all i where i(Xk)
appears for some Xk where k is not minimal among the Xi which appear. And
define the complication of a sum of monomials to be the greatest of their indiv*
*idual
complications. We may use our two families of relations to decrease complicatio*
*n,
which inductively allows us to reduce to our additive basis whose members_have
zero complication. |__|
Corollary 3.22.An additive basis for MOZ=2*is given by monomials of the form
ekoef, f 2 N*[Xi] and those of the form k(Xj)f, f 2 N*[Xi|i > j].
Proof of Theorem 2.6.Once again we use the fact that the map from MOZ=2*to
S-1MOZ=2*~=N*[xi; e; e-1] is a faithful representaion, along with direct comput*
*a-
tion. By Theorem 3.21, we may compute the image of [fl(M)] under localization
by analyzing fixed-set data.
Recalling the definition of fl(M) we see two types of fixed points [m; x; y] *
*under
the Z=2-action, those with x = 0 and those with y = 0 and gm = m. The first fix*
*ed
set is ff(M), with a trivial normal bundle. The second fixed set is the fixed s*
*et of
M, whose normal bundle is the normal bundle of this fixed set in M crossed with*
* a
REAL EQUIVARIANT BORDISM AND STABLE TRANSVERSALITY OBSTRUCTIONS FOR Z=211
trivial bundle. By the fact that multiplication by e-1 in N*[xi; e; e-1] corres*
*ponds
geometrically to crossing with a trivial bundle, this fixed set is the_fixed se*
*t of
([M]). |__|
Proof of Theorem 2.7.By analysis identical to that in the proof of Theorem 2.2,
the map ': NZ=2*! kN*-k(BO(k)) is injective.
We deduce from the comparison of exact sequences in Theorem 3.21 that the
Pontryagin-Thom map from NZ=2*to MOZ=2*is injective. So the image of ' is
the image of MOZ=2*in the subring N*[xi; e-1] of S-1MOZ=2*. The images of
[fli(P(no oe))] generate this image, so these classes generate NZ=2*. *
* |___|
Proof of Theorem 2.8.This theorem follows almost immediately from Corollary 3.22
and Theorem 2.7. Any monomial of the form k([P(jo oe)])f, f 2 N*[P(io oe)]|i >
j] is in fact in the image of the Pontryagin-Thom map. Monomials of the form ek*
*oef,
f 2 N*[P(io oe)] are generated over NZ=2*by ekoewhich we denote by xk. The
module relations for the quotient follow from the ring relations for MOZ=2*. *
* |___|
References
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* of the A.M.S.
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[2]G. Carlsson. A Survey of Equivariant Stable Homotopy Theory. Topology 31 (1*
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[3]P.E. Conner and E.E. Floyd. Differentiable Periodic Maps. Springer, Berlin-*
*Heidelberg-New
York, 1964.
[4]S. R. Costenoble and S. Waner. G-transversality revisited. Proceedings of t*
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[5]T. tom Dieck. Bordism of G-Manifolds and Integrality Theorems. Topology 9 (*
*1970), 345-358.
[6]J.P. May et al. Equivariant Homotopy and Cohomology Theory. Volume 91 of th*
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Regional Conference Series in Mathematics. AMS Publications, Providence, 199*
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[7]D. P. Sinha. Computations in Complex Equivariant Bordism Theory. Submitted *
*for publica-
tion.
Department of Mathematics, Brown University, Providence, RI 02906
E-mail address: dps@math.brown.edu