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What is an early reference for the fact that if a compact, connected $n$-manifold $M$ is covered by two open sets homeomorphic to $\mathbb{R}^n$ then $M$ is homeomorphic to $S^n$? And is it true that …

Read Chapter 4 of Thurston's notes http://library.msri.org/books/gt3m/. He produces infinitely many closed hyperbolic 3-manifold of different volumes, and hence non-homeomorphic, just by doing Dehn su …

Yes, $Y$ is still irreducible, and this holds by a simple connectivity argument. Suppose that $\Sigma \subset Y \subset X$ is a smoothly embedded 2-sphere. Since $X$ is irreducible, $\Sigma$ bounds …

Let $C \subset S^1 \subset \mathbb{R}^2$ be a Cantor set. Let $H_C$ be its convex hull, the smallest closed subset of $\mathbb{R}^2$ containing $C$. Then $H_C$ is not a manifold with corners. Its boun …

To address your second question, there is a simple construction of a Markov partition which inputs not "any topological partition" but which simply inputs the matrix $M \in SL_2(Z)$ that represents $T …