higher geometry / derived geometry
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Let be a 4-manifold which is connected and oriented.
The Pontryagin-Thom construction as above gives for the commuting diagram of sets
where denotes cohomotopy sets, denotes ordinary cohomology, denotes ordinary homology and is normally framed cobordism classes of normally framed submanifolds. Finally is the operation of pullback of the generating integral cohomology class on (by the nature of Eilenberg-MacLane spaces):
Now
, , are isomorphisms
is an isomorphism if is “odd” in that it contains at least one closed oriented surface of odd self-intersection, otherwise becomes an isomorphism on a -quotient group of (which is a group via the group-structure of the 3-sphere (SU(2)))
All PL 4-manifolds are simple branched covers of the 4-sphere:
Riccardo Piergallini, Four-manifolds as 4-fold branched covers of , Topology Volume 34, Issue 3, July 1995 (doi:10.1016/0040-9383(94)00034-I, pdf)
Massimiliano Iori, Riccardo Piergallini, 4-manifolds as covers of the 4-sphere branched over non-singular surfaces, Geom. Topol. 6 (2002) 393-401 (arXiv:math/0203087)
On cohomotopy of 4-manifolds:
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)
Robion Kirby, Paul Melvin, Peter Teichner, Cohomotopy sets of 4-manifolds, GTM 18 (2012) 161-190 (arXiv:1203.1608)
Last revised on December 20, 2020 at 22:02:20. See the history of this page for a list of all contributions to it.