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Every closed odd-dimensional manifold has Euler characteristic zero. For an orientable manifold this follows by pairing complementary Betti numbers using Poincare duality; the general case follows from the orientable double cover and Euler characteristic under a finite covering.

The standard CW complex structure on ${\mathbb{CP}}^n$ has one cell in every even dimension from $0$ to $2n$, and ${\mathbb{CP}}^k$ is its $2k$-skeleton. Collapsing that subcomplex leaves one zero-cell and one cell in each dimension $2i$ for $k+1\le i\le n$. All cellular boundaries vanish, so

Let $x\in H^2({\mathbb{CP}}^n;\mathbb{Z})$ be the usual generator. For $i>k$, choose $u_i$ whose pullback under the quotient map is $x^i$. Naturality of the cup product and the cohomology ring of complex projective space giveTogether with the unit, this determines the ring; equivalently, its reduced part is the ideal $(x^{k+1})/(x^{n+1})$ with the inherited multiplication. This is the cohomology ring of a collapsed projective subspace.

If $n=k+1$, the quotient has just a zero-cell and a $2n$-cell, so it is $S^{2n}$ and is a compact manifold. Conversely, suppose $n\ge k+2$ and the quotient is homotopy equivalent to a compact manifold. Its top cohomology is $\mathbb{Z}$, so that manifold must be closed, orientable, and $2n$-dimensional. But $H^{2n-2}\cong \mathbb{Z}$ while $H^2=0$, contradicting Poincare duality. Therefore

The cap product is the chain operationobtained by evaluating the cochain on the front $k$-face of a singular simplex and retaining its back $(p-k)$-face, with the standard sign convention. It descends to homology and cohomology.

A fundamental class $[M]\in H_d(M;\mathbb{Z})$ restricts at every $x\in M$ to the local generator of $H_d(M,M\setminus \{x\};\mathbb{Z})$ selected by the orientation. Poincare duality states thatis an isomorphism for every $k$ and coefficient ring $R$.

Let $0<k<d$ and $\alpha \in H^k(M;\mathbb{Z})$. Its restrictions to the contractible sets $U$ and $V$ vanish. The cap-product support lemma for a two-set cover, proved by representing $[M]$ with small simplices and replacing the restricted cocycles by coboundaries, therefore givesPoincare duality makes cap product with $[M]$ injective, so $H^k(M;\mathbb{Z})=0$ for every intermediate degree. Applying duality again gives $H_i(M;\mathbb{Z})=0$ for $0<i<d$. Finally the connected oriented closed manifold has