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Homotopy theory, homological algebra, algebraic treatments of manifolds.

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Identifying the orientation bundle uniquely

A nonorientable surface $S$ is homeomorphic to the $k$-th connected sum $\mathbb{R}P^2 \sharp \ldots \sharp \mathbb{R}P^2$. For each nonorientable surface $S$ there exists an oriented $2$-fold cov …