Relative mapping degree ($F_{\ast }[D^m,S^{m-1}]=d[D^m,S^{m-1}]$)

For a map of oriented disk pairs, the multiplier on top relative homology is its relative mapping degree. Naturality of the connecting homomorphism identifies it with the mapping degree of the boundary map.

For a smooth map of disk pairs whose graph of a function is transverse to the zero slice, orient the graph by its domain. With the tangent space of the zero slice ordered before that of the graph, the local smooth intersection number at a zero is $\mathrm{sgn}\det DF$. Summing these signs gives the relative mapping degree and hence the boundary mapping degree. Reversing the order of these two $m$-dimensional tangent spaces changes the sign by $(-1{)}^m$.

Collapsing the boundary identifies the top relative homology of a disk pair with the top reduced homology of its quotient sphere. Together with the boundary connecting homomorphism, this shows that the induced quotient map and the boundary map have the same mapping degree, using the corresponding orientations.