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Let $\sigma:S^3\times S^3\to S^3\times S^3$ such that $\sigma(a,b)=(ab,b^{-1})$. As $\sigma\circ\sigma$ is the identity map, $\sigma$ is a homeo of order $2$. The subset $X=S^3\times(S^3\setminus\{e\})$ is invariant under $\sigma$ and, if we call $G$ the group generated by $\sigma$ (which is cyclic of order two), you are asking for the cohomology of the ...


0

Since $\mathbb R^2 \times S^1$ is an open solid torus, if we remove one point, we have a space homotopy-equivalent to a torus with one meridional disk glued in: $(S^1 \times S^1) \cup (D^2 \times \{-1\})$. Pinching the disk to a point, this is a $S^2$ with two points identified. Expanding that point to an interval, yields the union of a sphere with an ...


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One just has to calculate. Observe that: Any morphism of discrete simplicial sets is a Kan fibration. The full subcategory of discrete simplicial sets is closed under limits. The relative matching objects are constructed using only (finite) limits. It follows that any morphism of discrete simplicial "spaces" is a Reedy fibration. In particular, since the ...



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