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If the suspension of $M$ is homotopy equivalent to a smooth manifold, does $M$ bound a contractible smooth manifold?

Here is a form of Whitehead's theorem that you can find, for instance, in Corollary 6.70 of Davis, James F.; Kirk, Paul, Lecture notes in algebraic topology, Graduate Studies in Mathematics. 35. ...
• 102k

Connectedness of a subspace of $M_2(\mathbb C)$

Your space is homeomorphic to $\Bbb C^*\times\Bbb C^3$. A product of connected spaces is connected.
• 39.8k
1 vote
Accepted

Continuous injection to manifold with boundary

From the fact that $U \subset M$ is open and that $f : U \to M$ is continuous, injective, and open, it follows that $V=f(U)$ is open in $M$ and that $f$ restricts to a homeomorphism $f : U \to V$. ...
• 124k
1 vote
Accepted

Locally compact metric space and separation property

Yes. Take $N = X \setminus U_2$. We have $U_1 \subset N$, thus $N$ is a closed neighborhood of $C$. The interior of $N$ is $\operatorname{int} N = X \setminus \overline U_2$ and the boundary of $N$ is ...
• 78.3k
1 vote

• 34.2k
1 vote

Kirby calculus on E8 plumbing

This is an old question, but in case anyone finds this helpful, here is a picture: Another way to view the handleslide: [As @Aru Ray pointed out, the mistake seems to be mixing half-twists and full ...
• 3,405

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