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The most basic nerve lemma can be found as Corollary 4G.3 in Hatcher's Algebraic Topology: If $\mathcal U$ is an open cover of a paracompact space $X$ such that every nonempty intersection of ...

Let $M$ be a closed, orientable, irreducible 3-manifold and having an infinite fundamental group. Is it true that $M$ admits $Sol$ geometry if and only if $\pi_1M$ is virtually solvable but not ...

Let $M$ be a smooth, compact manifold without a boundary, with its universal covering $\tilde{M} = \mathbb{R}^n$. If there exists an injective homomorphism $h: \pi_1(M) \rightarrow O(k) \ltimes \...

A popular way to describe a grope as the direct limit $L$ of a nested sequence of compact 2-dimensional polyhedra $L_0 \to L_1 \to L_2 \to \cdots$ obtained as follows. Take $L_0$ as some $S_g$, an ...