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Fold cobordisms and a Poincaré–Hopf-type theorem for the signature

### Boldizsár Kalmár

Algebraic & Geometric Topology 22 (2022) 2533–2586
DOI: 10.2140/agt.2022.22.2533
##### Abstract

We give complete geometric invariants of cobordisms of framed fold maps. These invariants are of two types. We take the immersion of the fold singular set into the target manifold together with information about nontriviality of the normal bundle of the singular set in the source manifold. These invariants were introduced in the author’s earlier works. We take the induced stable partial framing on the source manifold whose cobordisms were studied in general by Koschorke. We show that these invariants describe completely the cobordism groups of framed fold maps into ${ℝ}^{n}$. Then we look for dependencies between these invariants and we study fold maps of $4k$–dimensional manifolds into ${ℝ}^{2}$. We construct special fold maps, which are representatives of the fold cobordism classes and we also compute the cobordism groups. We obtain a Poincaré–Hopf-type formula, which connects local data of the singularities of a fold map of an oriented $4k$–dimensional manifold $M$ to the signature of $M$. We also study the unoriented case analogously and prove a similar formula about the twisting of the normal bundle of the fold singular set.

##### Keywords
fold singularity, fold map, signature, cobordism, stable framing, immersion
##### Mathematical Subject Classification 2010
Primary: 57R20, 57R45
Secondary: 57R25, 57R42, 57R70, 57R90
##### Publication
Revised: 26 July 2020
Accepted: 6 June 2021
Published: 13 December 2022
##### Authors
 Boldizsár Kalmár Alfréd Rényi Institute of Mathematics Hungarian Academy of Sciences Budapest Hungary