nLab equivariant bordism homology theory

Idea

The refinement of bordism homology theory to equivariant stable homotopy theory.

Related concepts

• equivariant bordism theory

• equivariant cobordism cohomology theory

flavors of bordism homology theories/cobordism cohomology theories, their representing Thom spectra and cobordism rings:

bordism theory$\;$M(B,f) (B-bordism):

• MFr

• MO, MSO, MSpin, MString, …

• MU, MSU, …

  MΩΩSU(n)

  MP, MR

• MSpin^c

• MSp

relative bordism theories:

• MOFr, MUFr, MSUFr

equivariant bordism theory:

• equivariant MFr

• equivariant MO

• equivariant MU

global equivariant bordism theory:

• global equivariant mO,

• global equivariant mU

algebraic:

• algebraic cobordism

References

Original discussion of equivariant cobordism classes of manifolds:

• Pierre Conner, Edwin Floyd, Differentiable periodic maps, Bull. Amer. Math. Soc. Volume 68, Number 2 (1962), 76-86 (euclid:bams/1183524501)

• Pierre Conner, Edwin Floyd, Maps of Odd Period, Annals of Mathematics, Second Series, Vol. 84, No. 2 (1966), pp. 132-156 (jstor:1970515)

• Robert Stong, Unoriented Bordism and Actions of Finite Groups, Memoirs AMS 103, American Mathematical Society, 1970 (ams:memo-1-103/4)

Original discussion of equivariant Thom spectra:

• Tammo tom Dieck, Kobordismentheorie und Transformationsgruppen, Aaarhus University Preprint Series 1968/69, No. 30 (gbooks)

• Tammo tom Dieck, Bordism of $G$-manifolds and integrality theorems, Topology Volume 9, Issue 4, November 1970, Pages 345-358 (doi:10.1016/0040-9383(70)90058-3)

Original discussion of the relation between equivariant bordism classes of manifolds and equivariant Thom spectra:

• Arthur Wasserman, section 3 of Equivariant differential topology, Topology Vol. 8, pp. 127-150, 1969 (pdf)

• Theodor Bröcker, Edward Hook, Stable equivariant bordism, Math Z 129, 269–277 (1972) (doi:10.1007/BF01187353)

• Edward Hook, Equivariant cobordism and duality, Trans. Amer. Math. Soc. 178 (1973), 241-258 (doi:10.1090/S0002-9947-1973-0321120-4)

• Tammo tom Dieck, Orbittypen und äquivariante Homologie II, Arch. Math. 26 (1975), no. 6, 650–662 (pdf)

• Bernhard Hanke, Geometric versus homotopy theoretic equivariant bordism, Math. Ann. 332, 677–696 (2005) (arXiv:math/0412550, doi:10.1007/s00208-005-0648-0)

On the Conner-Floyd isomorphism in equivariant cohomology theory, relating relating equivariant cobordism cohomology to equivariant K-theory:

• Steven Costenoble, Equivariant Conner-Floyd isomorphism, Trans. Amer. Math. Soc. 304 2 (1987) 801–818, [jstor:2000742]

Survey:

• Stefan Schwede, Equivariant bordism from the global perspective, 2015 (pdf, pdf)

• Michael Wiemeler, Equivariant bordism and applications in Differential Geometry, talk at Math Coll. Augsburg (Jan 2016) [pdf, pdf]

Discussion in global equivariant homotopy theory:

• Stefan Schwede, section 6.2 Global homotopy theory New Mathematical Monographs, 34 Cambridge University Press, 2018 (arXiv:1802.09382)

• Ningchuan Zhang, Equivariant cobordism (2019) [pdf]