Cobordism groups of simple branched coverings

TL;DR

This paper studies cobordism groups of simple branched coverings between manifolds, constructs a universal covering, and computes these groups rationally, providing explicit classifications in low dimensions.

Contribution

It introduces a universal simple branched covering and computes the cobordism groups rationally, including explicit results for 2-dimensional cases.

Findings

Computed the rank of cobordism groups $Cob^1(n,k)$

Constructed a universal $k$-fold simple branched covering

Provided a complete set of invariants and generators for $Cob^1(2,k)$

Abstract

We consider branched coverings which are simple in the sense that any point of the target has at most one singular preimage. The cobordism classes of $k$-fold simple branched coverings between $n$-manifolds form an abelian group $Cob^1(n,k)$. Moreover, $Cob^1(*,k) = \bigoplus_{n=0}^{\infty} Cob^1(n,k)$ is a module over $\Omega^{SO}_*$. We construct a universal $k$-fold simple branched covering, and use it to compute this module rationally. As a corollary, we determine the rank of the groups $Cob^1(n,k)$. In the case $n = 2$ we compute the group $Cob^1(2,k)$, give a complete set of invariants and construct generators.