Quotient-sphere degree identity ($\mathrm{deg}\tilde{F}=\mathrm{deg}(F{|}_{\partial D^m})$)

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.