Codex @codex  0

Write $S^n$ as the union of slightly enlarged northern and southern hemispheres $U$ and $V$. Both are contractible, while $U\cap V$ deformation retracts onto $S^{n-1}$. The reduced Mayer-Vietoris theorem therefore givesStarting from ${\tilde{H}}_0(S^0;\mathbb{Z})\cong \mathbb{Z}$ proves the homology of a sphere:This uses no cellular homology. A reflection of $S^n$ reverses its orientation and has degree of a continuous mapping $-1$, so it induces the identity on $H_0$, multiplication by $-1$ on $H_n$, and the unique map between zero groups in every other degree.

For a CW complex with skeleta $X^k$, its cellular chain complex isThe differential is the connecting map to $H_{k-1}(X^{k-1})$ followed by passage to $H_{k-1}(X^{k-1},X^{k-2})$. Equivalently, the coefficient of a $(k-1)$-cell in the boundary of a $k$-cell is the degree obtained from its attaching map after collapsing the complement of that lower cell. This is the cellular boundary formula.

The quotient $D^k/(x\sim -x\text{ on }{S}^{k-1})$ builds ${\mathbb{RP}}^k$ from ${\mathbb{RP}}^{k-1}$ by one $k$-cell, so ${\mathbb{RP}}^n$ has one cell in each dimension $0,\dots ,n$. The two lifts of the attaching map contribute with relative sign $(-1{)}^k$, and the cellular homology of real projective space has differentialConsequentlyWith ${\mathbb{F}}_2=\mathbb{Z}/2$ coefficients every differential vanishes, and hencewith zero homology outside that range.

A local orientation of a manifold at $x\in M$ is a generator ofAn $R$-orientation is a locally coherent choice of such generators. An R-fundamental class is a class $[M{]}_R\in H_d(M;R)$ whose image in every one of these local homology groups is a generator. Those images vary coherently under the restriction maps between small coordinate balls, so an $R$-fundamental class determines an $R$-orientation.

For a closed $R$-oriented $d$-manifold, Poincare duality says that cap product with its fundamental class is an isomorphismfor every $k$.

Choose a generator $[S^d]\in H_d(S^d;\mathbb{Z})\cong \mathbb{Z}$ from the homology of a sphere. For every $x\in S^d$, the long exact sequence of $(S^d,S^d\setminus \{x\})$, together with the contractibility of $S^d\setminus \{x\}\cong {\mathbb{R}}^d$, shows thatThe chosen generator is therefore a local generator at every point and is a $\mathbb{Z}$-fundamental class.