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 1. J.F. Adams, : :w:hat we don't know about RP1 , in New Developments in Topolo* *gy, ed. G. Segal, LMS Lecture Notes 11 (1972) 2. J.C. Alexander, The bordism ring of manifolds with involution, Proc. AMS 31 * *(1972) 536-542 3. J.M. Boardman, On manifolds with involution, BAMS 73 (1967) 136-138 4. _-, Cobordism of involutions revisited, Proc. Amherst conf. on transformatio* *n groups, Lec- ture notes in math. no. 298 (1971) 131-151 5. Th. Br"ocker, E.C. Hook, Stable equivariant bordism, Math. Zeits. 129 (1972)* * 269 - 277 6. M. Cole, J.P.C. Greenlees, I. Kriz, Equivariant formal group laws, * * posted on hopf.math.purdue.edu 7. P.E. Conner, E.E. Floyd, Differentiable periodic maps, Springer Ergebnisse 3* *3 (1964) 8. S. Costenoble, The structure of some equivariant Thom spectra, Trans. AMS 31* *5 (1989) 231-254 9. J.P.C. Greenlees, J.P. May, Generalized Tate cohomology, Mem. AMS 113 (1995) 10.I. Kriz, The Z=pZ-equivariant complex cobordism ring, these Proceedings 11.M. Mahowald, On the metastable homotopy of Sn, Mem. AMS 72 (1967) 12.D. Quillen, On the formal group laws of unoriented and complex cobordism the* *ory, BAMS 75 (1969) 1293-1298 13._-, Elementary proofs : :,:Adv. in Math. 7 (1971) 29-56 14.R. Swan, Periodic resolutions for finite groups, Annals of Math. 72 (1960) 2* *67-291 15.S. Waner, Equivariant RO(G)-graded bordism theories, Topology and its Applic* *ations 17 (1984) 1-26 Department of Mathematics, Johns Hopkins University, Baltimore, Maryland 212* *18 E-mail address: jack@math.jhu.edu