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Suppose $F:{\Sigma }_g\to {\Sigma }_h$ has degree one. For degree-one classes $u,v$ on the target, naturality of the cup product and the definition of the degree of a map between oriented manifolds giveIf a nonzero $u$ had $F^{\ast }u=0$, nondegeneracy of the Poincare duality pairing would supply $v$ with nonzero right side, a contradiction. Thus $F^{\ast }$ injects the $2h$-dimensional real degree-one cohomology into the $2g$-dimensional source. Hence $g\ge h$. This is the cohomological injectivity of a degree-one map in this setting.

Conversely, for $g\ge h$, express ${\Sigma }_g$ as ${\Sigma }_h\#{\Sigma }_{g-h}$. Collapse the second punctured summand and the joining circle to a point. The quotient of the retained punctured summand by its boundary is homeomorphic to ${\Sigma }_h$, giving a continuous map to that surface. Its restriction to a small oriented disc away from the collapsing region is an orientation-preserving homeomorphism, and a point in this disc has exactly one preimage. The induced map on local top homology, and hence on the fundamental class, has coefficient $+1$. The map therefore has degree one. For $h=0$, the same construction is the familiar collapse of the complement of a disc to obtain $S^2$.

ConsequentlyThis proves both directions of the degree-one maps between closed oriented surfaces criterion. The case $g=h$ also admits the identity map.

Part 1 makes $q:S^d\to N$ a degree-one map of closed oriented $d$-manifolds. The cohomological injectivity of a degree-one map embeds $H^{\ast }(N;\mathbb{Z})$ into $H^{\ast }(S^d;\mathbb{Z})$. Hence $N$ has no cohomology in degrees $0<i<d$; Poincare duality and the fundamental class giveThus $N$ is an integral homology sphere.

The long exact sequence of the pair $(N,N\setminus \{p\})$ has local relative homology $\mathbb{Z}$ only in degree $d$, and the map $H_d(N)\to H_d(N,N\setminus \{p\})$ is an isomorphism because it sends the fundamental class to the local orientation. Exactness now givesfor every $i$. Since $N\setminus \{p\}\cong S^d\setminus X$, the latter has the integral homology of a point.