Cobordism group of Morse functions on manifolds (Q1766194)

This paper defines and computes oriented and unoriented cobordism of manifolds with Morse functions. The equivalence relation assumes the cobordism is mapped into \(\mathbb{R}\times [0,1]\) with only fold singularities. The results say that the cobordism groups are direct sums of ordinary cobordism and copies of \(\mathbb{Z}\). The extra integer invariants are given by the indices of the critical points of the Morse function. For the oriented case of manifolds of dimension \(4k+1\), there is an extra invariant given by the difference between the semicharacteristic of \(M\) and a semicharacteristic computed from the Morse function.