cobordism theory = manifolds and cobordisms + stable homotopy theory/higher category theory
Concepts of cobordism theory
flavors of bordism homology theories/cobordism cohomology theories, their representing Thom spectra and cobordism rings:
relative bordism theories:
global equivariant bordism theory:
algebraic:
The first stable homotopy group of spheres (the first stable stem) is the cyclic group of order 2:
where the generator is represented by the complex Hopf fibration .
Under the Pontrjagin-Thom isomorphism, identifying the stable homotopy groups of spheres with the bordism ring of stably framed manifolds (see at MFr), the generator (1) is represented by the 1-sphere (with its left-invariant framing induced from the identification with the Lie group U(1))
Moreover, the relation is represented by the bordism which is the complement of 2 open balls inside the 2-sphere.
The original computation via Pontryagin's theorem in cobordism theory:
with a more comprehensive account in:
Review:
Daniel Freed, Karen Uhlenbeck, Appendix B of: Instantons and Four-Manifolds, Mathematical Sciences Research Institute Publications, Springer 1991 (doi:10.1007/978-1-4613-9703-8)
Guozhen Wang, Zhouli Xu, Section 2.3 of: A survey of computations of homotopy groups of Spheres and Cobordisms, 2010 (pdf)
Andrew Putman, Section 5 of: Homotopy groups of spheres and low-dimensional topology (pdf, pdf)
Last revised on February 4, 2021 at 06:23:05. See the history of this page for a list of all contributions to it.