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:
By one denotes the Thom spectrum for stable Sp(n)-structure, representing what is called “symplectic” or “quaternionic” cobordism cohomology theory. The coefficient ring is, accordingly, the “quaternionic”/“symplectic” bordism ring, usually denoted .
The canonical topological group-inclusions
(trivial group into quaternionic unitary group into special unitary group into unitary group) induce ring spectrum-homomorphism of Thom spectra
(from MFr to MSp to MSU to MU)
and hence corresponding multiplicative cohomology theory-homomorphisms of cobordism cohomology theories.
(e.g. Conner-Floyd 66, p. 27 (34 of 120))
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:
Sergei Novikov, Homotopy properties of Thom complexes, Mat. Sbornik 57 (1962), no. 4, 407–442, 407–442 (pdf, pdf)
Arunas Liulevicius, Notes on Homotopy of Thom Spectra, American Journal of Mathematics Vol. 86, No. 1 (1964), pp. 1-16 (jstor:2373032)
Pierre Conner, Edwin Floyd, Section 5 of: The Relation of Cobordism to K-Theories, Lecture Notes in Mathematics 28 Springer 1966 (doi:10.1007/BFb0071091, MR216511)
Robert Stong, Some Remarks on Symplectic Cobordism, Annals of Mathematics Second Series, Vol. 86, No. 3 (Nov., 1967), pp. 425-433 (doi:10.2307/1970608)
D. M. Segal, On the symplectic cobordism ring, Commentarii Mathematici Helvetici 45, 159–169 (1970) (doi:10.1007/BF02567323)
Nigel Ray, The symplectic bordism ring, Volume 71, Issue 2 March 1972, pp. 271-282 (doi:10.1017/S0305004100050519)
Stanley Kochman, The symplectic cobordism ring I, Memoirs of the American Mathematical Society 1980, Volume 24, Number 271 (doi:10.1090/memo/0271)
Stanley Kochman, The symplectic cobordism ring II, Memoirs of the American Mathematical Society 1982
Volume 40, Number 271 (doi:10.1090/memo/0271)
Andrew Baker, Some chromatic phenomena in the homotopy of , in: N. Ray, G. Walker (eds.), Adams Memorial Symposium on Algebraic Topology, Vol. 2 editors, Cambridge University Press (1992), 263–80 (pdf, pdf)
Vassily Gorbounov, Nigel Ray, Orientations of Bundles and Symplectic Cobordism, Publ. RIMS, Kyoto Univ. 28 (1992), 39-55 (pdf, doi: 10.2977/prims/1195168855)
Vassily Gorbounov, Nigel Ray, Paul Turner, On the Hopf Ring for a Symplectic Oriented Spectrum, American Journal of Mathematics
Vol. 117, No. 4 (Aug., 1995), pp. 1063-1088 (doi:10.2307/2374960)
Craig Laughton, Quasitoric manifolds and cobordism theory, Manchester 2008 (pdf, pdf)
Ivan Panin, Charles Walter, On the algebraic cobordism spectra and (arXiv:1011.0651)
Last revised on February 10, 2024 at 09:49:09. See the history of this page for a list of all contributions to it.