## Algebraic & Geometric Topology

### Equivariant cohomology Chern numbers determine equivariant unitary bordism for torus groups

#### Abstract

We show that the integral equivariant cohomology Chern numbers completely determine the equivariant geometric unitary bordism classes of closed unitary $G$–manifolds, which gives an affirmative answer to a conjecture posed by Guillemin, Ginzburg and Karshon (Moment maps, cobordisms, and Hamiltonian group actions, Remark H.5 in Appendix H.3), where $G$ is a torus. As a further application, we also obtain a satisfactory solution of their Question (A) (Appendix H.1.1) on unitary Hamiltonian $G$–manifolds. Our key ingredients in the proof are the universal toric genus defined by Buchstaber, Panov and Ray and the Kronecker pairing of bordism and cobordism. Our approach heavily exploits Quillen’s geometric interpretation of homotopic unitary cobordism theory. Moreover, this method can also be applied to the study of $( ℤ 2 ) k$–equivariant unoriented bordism and can still derive the classical result of tom Dieck.

#### Article information

Source
Algebr. Geom. Topol., Volume 18, Number 7 (2018), 4143-4160.

Dates
Revised: 23 April 2018
Accepted: 10 June 2018
First available in Project Euclid: 18 December 2018

https://projecteuclid.org/euclid.agt/1545102068

Digital Object Identifier
doi:10.2140/agt.2018.18.4143

Mathematical Reviews number (MathSciNet)
MR3892242

Zentralblatt MATH identifier
07006388

#### Citation

Lü, Zhi; Wang, Wei. Equivariant cohomology Chern numbers determine equivariant unitary bordism for torus groups. Algebr. Geom. Topol. 18 (2018), no. 7, 4143--4160. doi:10.2140/agt.2018.18.4143. https://projecteuclid.org/euclid.agt/1545102068

#### References

• J F Adams, Stable homotopy and generalised homology, Univ. Chicago Press (1974)
• M F Atiyah, Bordism and cobordism, Proc. Cambridge Philos. Soc. 57 (1961) 200–208
• M F Atiyah, R Bott, A Lefschetz fixed point formula for elliptic complexes, II: Applications, Ann. of Math. 88 (1968) 451–491
• M F Atiyah, G B Segal, The index of elliptic operators, II, Ann. of Math. 87 (1968) 531–545
• M F Atiyah, I M Singer, The index of elliptic operators, I, Ann. of Math. 87 (1968) 484–530
• M Bix, T tom Dieck, Characteristic numbers of $G$–manifolds and multiplicative induction, Trans. Amer. Math. Soc. 235 (1978) 331–343
• J M Boardman, Stable homotopy theory, VI, mimeographed notes, University of Warwick (1966)
• V M Buchstaber, T E Panov, Toric topology, Mathematical Surveys and Monographs 204, Amer. Math. Soc., Providence, RI (2015)
• V Buchstaber, T Panov, N Ray, Toric genera, Int. Math. Res. Not. 2010 (2010) 3207–3262
• G Comezaña, Calculations in complex equivariant bordism, from “Equivariant homotopy and cohomology theory” (J P May, editor), CBMS Regional Conference Series in Mathematics 91, Amer. Math. Soc., Providence, RI (1996) 333–352
• P E Conner, Seminar on periodic maps, Lecture Notes in Mathematics 46, Springer (1967)
• P E Conner, E E Floyd, Differiable periodic maps, Bull. Amer. Math. Soc. 68 (1962) 76–86
• P E Conner, E E Floyd, Differentiable periodic maps, Ergeb. Math. Grenzgeb. 33, Springer (1964)
• P E Conner, E E Floyd, Periodic maps which preserve a complex structure, Bull. Amer. Math. Soc. 70 (1964) 574–579
• T tom Dieck, Bordism of $G$–manifolds and integrality theorems, Topology 9 (1970) 345–358
• T tom Dieck, Characteristic numbers of $G$–manifolds, I, Invent. Math. 13 (1971) 213–224
• T tom Dieck, Characteristic numbers of $G$–manifolds, II, J. Pure Appl. Algebra 4 (1974) 31–39
• V Guillemin, V Ginzburg, Y Karshon, Moment maps, cobordisms, and Hamiltonian group actions, Mathematical Surveys and Monographs 98, Amer. Math. Soc., Providence, RI (2002)
• B Hanke, Geometric versus homotopy theoretic equivariant bordism, Math. Ann. 332 (2005) 677–696
• A Hattori, Equivariant characteristic numbers and integrality theorem for unitary $T\sp{n}$–manifolds, Tôhoku Math. J. 26 (1974) 461–482
• S S Khare, Characteristic numbers for oriented singular $G$–bordism, Indian J. Pure Appl. Math. 13 (1982) 637–642
• S O Kochman, Bordism, stable homotopy and Adams spectral sequences, Fields Institute Monographs 7, Amer. Math. Soc., Providence, RI (1996)
• I M Krichever, Obstructions to the existence of $S\sp{1}$–actions: bordisms of branched coverings, Izv. Akad. Nauk SSSR Ser. Mat. 40 (1976) 828–844 In Russian; translated in Math. USSR-Izv. 10 (1976) 783–797
• P S Landweber, A survey of bordism and cobordism, Math. Proc. Cambridge Philos. Soc. 100 (1986) 207–223
• C N Lee, A G Wasserman, Equivariant characteristic numbers, from “Proceedings of the second conference on compact transformation groups, I” (H T Ku, L N Mann, J L Sicks, J C Su, editors), Lecture Notes in Math. 298, Springer (1972) 191–216
• P Löffler, Bordismengruppen unitärer Torusmannigfaltigkeiten, Manuscripta Math. 12 (1974) 307–327
• Z Lü, Q Tan, Equivariant Chern numbers and the number of fixed points for unitary torus manifolds, Math. Res. Lett. 18 (2011) 1319–1325
• Z Lü, Q Tan, Small covers and the equivariant bordism classification of $2$–torus manifolds, Int. Math. Res. Not. 2014 (2014) 6756–6797
• J Milnor, On the cobordism ring $\Omega \sp{\ast}$ and a complex analogue, I, Amer. J. Math. 82 (1960) 505–521
• S P Novikov, Some problems in the topology of manifolds connected with the theory of Thom spaces, Dokl. Akad. Nauk SSSR 132 (1960) 1031–1034 In Russian; translated in Soviet Math. Dokl. 1 (1960) 717–720
• D Quillen, Elementary proofs of some results of cobordism theory using Steenrod operations, Advances in Math. 7 (1971) 29–56
• D P Sinha, Computations of complex equivariant bordism rings, Amer. J. Math. 123 (2001) 577–605
• R E Stong, Notes on cobordism theory, Mathematical Notes 7, Princeton Univ. Press (1968)
• R E Stong, Equivariant bordism and Smith theory, IV, Trans. Amer. Math. Soc. 215 (1976) 313–321
• R Thom, Quelques propriétés globales des variétés différentiables, Comment. Math. Helv. 28 (1954) 17–86
• C T C Wall, Determination of the cobordism ring, Ann. of Math. 72 (1960) 292–311