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It is well-known that every closed, connected and orientable 3-manifold $\mathcal{M}$ can uniquely be decomposed as $$\mathcal{M}=P_{1}\#\dots\# P_{n}$$ where $P_{i}$ are prime manifolds, i.e. ...

Let $n\ge 3$ and $X$ be a compact connected $n$-manifold (without boundary). I need a reference to the following facts (which I believe are true at least in dimension $n=3$): Fact 1. For every ...

It seems to be widely known that every topological manifold can be embedded as a neighbourhood retract in euclidean space, I can not find a reference, though. The reason, why I'm asking this, is that ...

Let $M$ be a (not necessarily Hausdorff) smooth manifold. Given an open chart $U\subset M$ and a compactly-supported smooth function $f:U\to\mathbb{R}$ on $U$, define $\widetilde{f}:M\to\mathbb{R}$ by ...