Cobordism of involutions revisited, revisited
Jack Morava
Abstract.Boardman's work [3,4] on Conner and Floyd's five-halves conjec-
ture looks remarkably contemporary in retrospect. This note reexamines s*
*ome
of that work from a perspective proposed recently by Greenlees and Kriz.
1. Unoriented involutions
A large part of Boardman's argument can be summarized as a commutative diagram
0_____//NgeoZ=2Z*_//i0 Ni(BO*-i)__@__//eN*-1(BZ=2Z)____//_0
| | |
| |J |
fflffl| fflffl| fflffl|
0_____//N*[[w]]______//N*((w))________//_N*-1[w-1]_____//_0 :
of short exact sequences: here N*((w)) is the ring of homogeneous formal Laurent
series over the unoriented cobordism ring, in an indeterminate w of cohomologic*
*al
degree one; such series are permitted only finitely many negative powers of w.
The lower left-hand arrow is inclusion of the ring of formal power series, and *
*the
lower right-hand arrow is the quotient homomorphism. The maps across the top
are geometric: the left-hand arrow sends the class of an unoriented manifold wi*
*th
involution to its class in the relative bordism of unoriented Z=2Z-manifolds-wi*
*th-
boundary with action free on the boundary, while the right-hand arrow @ sends
such a manifold to its boundary, regarded as a manifold with involution classif*
*ied
by a map to BZ=2Z. It is a theorem of Conner and Floyd [7 x22] that such a rela*
*tive
manifold with involution is cobordant to the normal ball bundle of its fixed-po*
*int
set, so the group in the top row center is the sum of bordism groups of classif*
*ying
spaces for orthogonal groups, as indicated.
Remarks:
i) Thom showed that unoriented cobordism is a generalized Eilenberg-Mac Lane
spectrum, but this is not true of equivariant unoriented bordism, cf. [8];
ii) the homomorphism @ is not a derivation;
____________
1991 Mathematics Subject Classification. Primary 55-03, Secondary 55N22, 55*
*N91.
The author was supported in part by the NSF.
1
2 JACK MORAVA
iii) the right-hand vertical arrow is an isomorphism, while the left-hand ve*
*rtical
arrow sends a closed manifold with involution V to the bordism class of its Bor*
*el
construction, regarded as an element of N-*(BZ=2Z), cf. [13];
iv) the middle vertical homomorphism J is a ring homomorphism; its construc-
tion [4, Theorem 9] is one of Boardman's innovations. He concludes that it is i*
*n fact
a monomorphism [4, Corollary 17], so NgeoZ=2Z*might be described as the subring*
* of
classes [V ] in the relative bordism group such that J[V ] is `holomorphic' in *
*w. The
homomorphism J is constructed using an inverse system of truncated projective
spaces; around the same time Mahowald [1,11] called attention to the remarkable
properties of a similar construction in framed bordism, and went on to consider
the properties of this inverse limit : : :
v) The class of the interval [-1; +1] (with sign reversal as involution) def*
*ines
an element of the relative bordism group which maps under J to w-1, cf. [5].
In modern terminology the diagram displays the Tate Z=2Z-cohomology [[9]; Swan
probably also deserves mention here, cf. [14]] of the forgetful map from geomet*
*ric
Z=2Z-equivariant unoriented cobordism to ordinary unoriented cobordism; this is
very closely related to the construction used by Kriz [10 Corollary 1.4] to com*
*pute
the homotopy-theoretic Z=pZ-equivariant bordism groups MUhotZ=pZ*. In fact the
direct limit of the system defined by suspending the diagram with respect to a
complete family of Z=2Z-representations (cf. [15]) has top row
0 ! NhotZ=2Z*! N*(BO)[u; u-1] ! eN*(BZ=2Z) ! 0 ;
which is identical [aside from obvious modifications] with Kriz's. It follows,*
* in
particular, that the stabilization map
NgeoZ=2Z*! NhotZ=2Z*
is injective.
2. Complex circle actions
Boardman has an explicit formula for his J-homomorphism: if [V ] is the cobordi*
*sm
class of a closed manifold with involution, then
X
J[V ] = ss-1 wkp*@(w-k-1V ) ;
k0
where p* : N*(BZ=2Z) ! N* sends a manifold with free involution to its quotient,
and X
ss := [RPk]wk ;
k0
by [4 Theorem 14]. This formula has a natural interpretation in terms of formal
group laws, but the idea is easier to explain in the context of the T-equivaria*
*nt Tate
cohomology of MU, which fits in an exact sequence
0 ! MU-* (BT) ! tT(MU*) = MU*((c)) ! MU*-1(BT) ! 0 ;
here the Chern class c plays the role of the Stiefel-Whitney class w in the uno*
*riented
case. Since the complex cobordism ring is torsion-free, we can introduce denomi-
nators with impunity: the analogue of w-1 is the class c-1 of the two-disk, vie*
*wed
COBORDISM OF INVOLUTIONS REVISITED, REVISITED 3
as a complex-oriented manifold with circle action free on the boundary. Boardman
observes that
p*@w-k-1 = [RPk] ;
similarly, we have
p*@c-k-1 = [CPk]
in the complex-oriented situation.
Proposition: The homomorphism p*@ is the formal residue at the origin with
respect to an additive coordinate for the formal group law of MU.
Proof: The idea is to write c as a formal power series
c = z + terms of higher order;
with coefficients chosen so that
resz=0c-k-1 = [CPk] :
Clearly
X c-1 X
res0c-k-1uk = res0_______-1= [CPk]uk = log0MU(u) :
k0 1 - c u k0
On the other hand, if c = f(z) then
Z
res0(f(z) - u)-1 = _1_2ssi_dz____f(z) - u
which equals
_1_ Z (f-1_)0(c)dc_= (f-1 )0(u) ;
2ssi c - u
thus c = expMU (z).
Remarks:
i) In the context of remark iv) above, Boardman's formula for J on `holomor-
phic' elements V is thus essentially the same as Cauchy's. His use of the symbo*
*l ss
suggests that this analogy was not far from his mind.
ii) The formal residue was introduced in Quillen's Bulletin announcement [12*
*],
but seems to have since disappeared from algebraic topology. Perhaps that paper
deserves another look as well.
iii) These constructions suggest that while the Chern class c is a natural u*
*ni-
formizing parameter for algebraic questions about complex cobordism, its inverse
may be more natural for geometric questions. We thus have two reasonable coordi-
nates on cobordism, centered at the south and north poles of the Riemann sphere,
overlapping in the temperate region defined by Tate cohomology.
Acknowledgements: I would like to thank John Greenlees, Igor Kriz, and Dev
Sinha for raising my consciousness about equivariant cobordism, and Bob Stong f*
*or
helpful conversations about the history of this subject.
4 JACK MORAVA
References
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*gy, ed. G.
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Department of Mathematics, Johns Hopkins University, Baltimore, Maryland 212*
*18
E-mail address: jack@math.jhu.edu