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Choose an oriented embedded closed $d$-disc $D$ in $M$ and collapse $M\setminus \mathrm{int}D$ to a point. The quotient is $D/\partial D\cong S^d$, with orientation chosen to agree on the disc interior. The resulting collapse map of degree one onto a sphere isIndeed, any target point other than the collapsed point has exactly one preimage, where the map is an orientation-preserving homeomorphism and has local degree $+1$. The degree as a sum of local degrees therefore gives global degree one. This is the usual connected, positive-dimensional setting for mapping degree; extra components, if allowed, may all be collapsed.