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Let $M$ be a compact oriented $n$-manifold. Denote $\Omega^k := {\bigwedge}^k T^*M$ the vector bundle of differential $k$-forms, and let $\Omega^{\text{odd}} := \bigoplus_{\text{$k$ odd}} \Omega^k$ ...

Let $(Z,A)$ a compact ENR pair, then $$\chi(Z)=\chi_c(Z-A)+\chi(A)$$ where $\chi_c$ is the Euler characteristic taken in Alexander-Spanier cohomology with compact support (ENR means euclidean ...

Edit: According to the interesting comments of Michael Albanese and Nick L we revise the question as follows: By manifold compactification of a manifold $M$ we mean a compact manifold $\tilde{M}$ ...