= Homology after attaching a Möbius band to the real projective plane
{title2=$H_1\cong\mathbb Z/4\text{ or }(\mathbb Z/2)^2$}
Glue the boundary of a <Möbius band> to the projective line in the <real projective plane> by a <mapping degree> $n$. The band is the <mapping cylinder> of the degree-two boundary covering of its core. Collapsing the connecting tree in the resulting double mapping cylinder gives a <CW complex> with one vertex, edges $a,b$, and two attaching words $a^2$ and $a^{-n}b^2$. Its <cellular chain complex> has degree-two boundary matrix $\left(\begin{smallmatrix}2&-n\\0&2\end{smallmatrix}\right)$. Its determinant is four, so $H_2=0$. The <Smith normal form> gives $H_1\cong\mathbb Z/4$ for odd $n$ and $(\mathbb Z/2)^2$ for even $n$; $H_0=\mathbb Z$ and higher <homology> vanishes. The space cannot be <homotopy equivalent> to a <closed manifold>: mod-two <homology> forces a potential manifold to be a surface, and its <Euler characteristic> is one, whereas the only such closed connected surface is the projective plane, with different integral $H_1$.
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