cobordism theory = manifolds and cobordisms + stable homotopy theory/higher category theory
Concepts of cobordism theory
homotopy classes of maps to Thom space MO
complex cobordism cohomology theory
flavors of bordism homology theories/cobordism cohomology theories, their representing Thom spectra and cobordism rings:
bordism theoryM(B,f) (B-bordism):
relative bordism theories:
global equivariant bordism theory:
algebraic:
The lift of complex cobordism cohomology theory to equivariant stable homotopy theory.
For an abelian compact Lie group , equivariant complex cobordism theory is an equivariant complex oriented cohomology theory (Greenlees 01, Sec. 13).
Much as in the non-equivariant case (see at universal complex orientation on MU), is universal in that there is a bijection between equivariant complex orientations (in degree 2) on some cohomology theory and homotopy ring homomorphisms of -spectra (Cole-Greenlees-Kriz 02, Theorem 1.2).
For the analogous statement on the equivariant Lazard ring see Greenlees 01a, Greenlees 01, Theorem 13.1, Cole-Greenlees-Kriz 02, Theorem 1.3.
flavors of bordism homology theories/cobordism cohomology theories, their representing Thom spectra and cobordism rings:
bordism theoryM(B,f) (B-bordism):
relative bordism theories:
global equivariant bordism theory:
algebraic:
Peter May, Equivariant and non-equivariant module spectra, Journal of Pure and Applied Algebra Volume 127, Issue 1, 1 May 1998, Pages 83–97 (pdf)
John Greenlees, The coefficient ring of equivariant homotopical bordism classifies equivariant formal group laws over Noetherian rings, 2001 (preprint)
John Greenlees, Equivariant formal group laws and complex oriented cohomology theories, Homology Homotopy Appl. Volume 3, Number 2 (2001), 225-263 (euclid:hha/1139840255)
Michael Cole, John Greenlees, Igor Kriz, The universality of equivariant complex bordism, Math Z 239, 455–475 (2002) (doi:10.1007/s002090100315)
William Abram, A note on the equivariant formal group law of the equivariant complex cobordism ring (arXiv:1309.0722)
William Abram, Igor Kriz, The equivariant complex cobordism ring of a finite abelian group (arXiv:1509.08540)
Generalization of the Conner-Floyd isomorphism to equivariant cohomology theory, relating equivariant cobordism cohomology to equivariant K-theory:
Last revised on September 17, 2022 at 09:41:02. See the history of this page for a list of all contributions to it.