aa r X i v : . [ m a t h . A T ] J un Cobordism of involutions revisited, revisited
Jack Morava
Abstract.
Boardman’s work [3,4] on Conner and Floyd’s five-halvesconjecture looks remarkably contemporary in retrospect. This note re-examines some of that work from a perspective proposed recently byGreenlees and Kriz.
1. Unoriented involutions
A large part of Boardman’s argument can be summarized as a commutativediagram0 / / N geo Z / Z ∗ / / (cid:15) (cid:15) ⊕ i ≥ N i ( BO ∗− i ) ∂ / / J (cid:15) (cid:15) e N ∗− ( B Z / Z ) / / (cid:15) (cid:15) / / N ∗ [[ w ]] / / N ∗ (( w )) / / N ∗− [ w − ] / / . of short exact sequences: here N ∗ (( w )) is the ring of homogeneous formalLaurent series over the unoriented cobordism ring, in an indeterminate w of cohomological degree one; such series are permitted only finitely manynegative powers of w . The lower left-hand arrow is inclusion of the ringof formal power series, and the lower right-hand arrow is the quotient ho-momorphism. The maps across the top are geometric: the left-hand arrowsends the class of an unoriented manifold with involution to its class in therelative bordism of unoriented Z / Z -manifolds-with-boundary with actionfree on the boundary, while the right-hand arrow ∂ sends such a manifoldto its boundary, regarded as a manifold with involution classified by a mapto B Z / Z . It is a theorem of Conner and Floyd [7 §
22] that such a relativemanifold with involution is cobordant to the normal ball bundle of its fixed-point set, so the group in the top row center is the sum of bordism groupsof classifying spaces for orthogonal groups, as indicated.
Mathematics Subject Classification.
Primary 55-03, Secondary 55N22, 55N91.The author was supported in part by the NSF.
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;iii) the right-hand vertical arrow is an isomorphism, while the left-handvertical arrow sends a closed manifold with involution V to the bordismclass of its Borel construction, regarded as an element of N −∗ ( B Z / Z ), cf.[13];iv) the middle vertical homomorphism J is a ring homomorphism; itsconstruction [4, Theorem 9] is one of Boardman’s innovations. He concludesthat it is in fact a monomorphism [4, Corollary 17], so N geo Z / Z ∗ might bedescribed as the subring of classes [ V ] in the relative bordism group suchthat J [ V ] is ‘holomorphic’ in w . The homomorphism J is constructed usingan inverse system of truncated projective spaces; around the same timeMahowald [1,11] called attention to the remarkable properties of a similarconstruction in framed bordism, and went on to consider the properties ofthis inverse limit . . .v) The class of the interval [ − , +1] (with sign reversal as involution)defines an element of the relative bordism group which maps under J to w − , cf. [5].In modern terminology the diagram displays the Tate Z / Z -cohomology [[9];Swan probably also deserves mention here, cf. [14]] of the forgetful map fromgeometric Z / Z -equivariant unoriented cobordism to ordinary unorientedcobordism; this is very closely related to the construction used by Kriz [10Corollary 1.4] to compute the homotopy-theoretic Z /p Z -equivariant bordismgroups M U hot Z /p Z ∗ . In fact the direct limit of the system defined by suspendingthe diagram with respect to a complete family of Z / Z -representations (cf.[15]) has top row0 → N hot Z / Z ∗ → N ∗ ( BO )[ u, u − ] → e N ∗ ( B Z / Z ) → , which is identical [aside from obvious modifications] with Kriz’s. It follows,in particular, that the stabilization map N geo Z / Z ∗ → N hot Z / Z ∗ is injective.
2. Complex circle actions
Boardman has an explicit formula for his J -homomorphism: if [ V ] is thecobordism class of a closed manifold with involution, then J [ V ] = π − X k ≥ w k p ∗ ∂ ( w − k − V ) , OBORDISM OF INVOLUTIONS REVISITED, REVISITED 3 where p ∗ : N ∗ ( B Z / Z ) → N ∗ sends a manifold with free involution to itsquotient, and π := X k ≥ [ R P k ] w k , by [4 Theorem 14]. This formula has a natural interpretation in terms offormal group laws, but the idea is easier to explain in the context of the T -equivariant Tate cohomology of M U , which fits in an exact sequence0 → M U −∗ ( B T ) → t T ( M U ∗ ) = M U ∗ (( c )) → M U ∗− ( B T ) → c plays the role of the Stiefel-Whitney class w in theunoriented case. Since the complex cobordism ring is torsion-free, we canintroduce denominators with impunity: the analogue of w − is the class c − of the two-disk, viewed as a complex-oriented manifold with circle actionfree on the boundary. Boardman observes that p ∗ ∂w − k − = [ R P k ] ;similarly, we have p ∗ ∂c − k − = [ C P k ]in the complex-oriented situation. Proposition:
The homomorphism p ∗ ∂ is the formal residue at the originwith respect to an additive coordinate for the formal group law of M U . Proof:
The idea is to write c as a formal power series c = z + terms of higher order , with coefficients chosen so thatres z =0 c − k − = [ C P k ] . Clearly X k ≥ res c − k − u k = res c − − c − u = X k ≥ [ C P k ] u k = log ′ MU ( u ) . On the other hand, if c = f ( z ) thenres ( f ( z ) − u ) − = 12 πi Z dzf ( z ) − u which equals 12 πi Z ( f − ) ′ ( c ) dcc − u = ( f − ) ′ ( u ) ;thus c = exp MU ( z ). Remarks: i) In the context of remark iv) above, Boardman’s formula for J on‘holomorphic’ elements V is thus essentially the same as Cauchy’s. His useof the symbol π suggests that this analogy was not far from his mind. JACK MORAVA ii) The formal residue was introduced in Quillen’s Bulletin announce-ment [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 naturaluniformizing parameter for algebraic questions about complex cobordism,its inverse may be more natural for geometric questions. We thus have tworeasonable coordinates on cobordism, centered at the south and north polesof the Riemann sphere, overlapping in the temperate region defined by Tatecohomology. Acknowledgements:
I would like to thank John Greenlees, Igor Kriz,and Dev Sinha for raising my consciousness about equivariant cobordism,and Bob Stong for helpful conversations about the history of this subject.
Postscript:
This has appeared in the Boardman Festschrift
Homotopy invariant algebraic structures
15 - 18, Contemp. Math.239, AMS (1999).I am posting it here in hopes of advertising this geometric description ofTate cohomology. Similar ideas play a role inHeisenberg groups and algebraic topology, in the Segal Festscrift
Topology,geometry and quantum field theory
235 - 246, LMS Lecture Notes 308,Cambridge (2004); as well as inCompletions of Z / ( p )-Tate cohomology of periodic spectra. Geom. Topol.2 (1998) 145 - 174, arXiv:math/9808141 OBORDISM OF INVOLUTIONS REVISITED, REVISITED 5
References
1. J.F. Adams, . . . what we don’t know about R P ∞ , in New Developments in Topology,ed. G. Segal, LMS Lecture Notes 11 (1972)2. J.C. Alexander, The bordism ring of manifolds with involution, Proc. AMS 31 (1972)536-5423. J.M. Boardman, On manifolds with involution, BAMS 73 (1967) 136-1384. —–, Cobordism of involutions revisited, Proc. Amherst conf. on transformationgroups, Lecture notes in math. no. 298 (1971) 131-1515. Th. Br¨ocker, E.C. Hook, Stable equivariant bordism, Math. Zeits. 129 (1972) 269 -2776. 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 33(1964)8. S. Costenoble, The structure of some equivariant Thom spectra, Trans. AMS 315(1989) 231-2549. J.P.C. Greenlees, J.P. May,
Generalized Tate cohomology , Mem. AMS 113 (1995)10. I. Kriz, The Z /p Z -equivariant complex cobordism ring, these Proceedings11. M. Mahowald, On the metastable homotopy of S n , Mem. AMS 72 (1967)12. D. Quillen, On the formal group laws of unoriented and complex cobordism theory,BAMS 75 (1969) 1293-129813. —–, Elementary proofs . . . , Adv. in Math. 7 (1971) 29-5614. R. Swan, Periodic resolutions for finite groups, Annals of Math. 72 (1960) 267-29115. S. Waner, Equivariant RO ( G )-graded bordism theories, Topology and its Applications17 (1984) 1-26 Department of Mathematics, Johns Hopkins University, Baltimore, Mary-land 21218
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