Fold maps, framed immersions and smooth structures
Abstract
For each integer q>0 there is a cohomology theory such that the zero cohomology group of a manifold N of dimension n is a certain group of cobordism classes of proper fold maps of manifolds of dimension n+q into N. We prove a splitting theorem for the spectrum representing the cohomology theory of fold maps. For even q, the splitting theorem implies that the cobordism group of fold maps to a manifold N is a sum of q/2 cobordism groups of framed immersions to N and a group related to diffeomorphism groups of manifolds of dimension q+1. Similarly, in the case of odd q, the cobordism group of fold maps splits off (q-1)/2 cobordism groups of framed immersions. The proof of the splitting theorem gives a partial splitting of the homotopy cofiber sequence of Thom spectra in the Madsen-Weiss approach to diffeomorphism groups of manifolds.
- Publication:
-
arXiv e-prints
- Pub Date:
- March 2008
- DOI:
- 10.48550/arXiv.0803.3780
- arXiv:
- arXiv:0803.3780
- Bibcode:
- 2008arXiv0803.3780S
- Keywords:
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- Mathematics - Geometric Topology;
- Mathematics - Algebraic Topology;
- 58K30;
- 55P42
- E-Print:
- 23 pages