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4 votes
Accepted

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. ...
Moishe Kohan's user avatar
4 votes

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.
Anne Bauval's user avatar
  • 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 $f : U \to V$. ...
Lee Mosher's user avatar
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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 ...
Paul Frost's user avatar
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1 vote

If a Banach manifold satisfies the Heine-Borel property, then does it have finite dimension?

Depending on your definition of "manifold", the Hilbert cube may be a counterexample. The Hilbert cube is the subset of $\ell^2$ consisting of those sequences $(x_n)$ satisfying $0\le x_n \...
Giuseppe Negro's user avatar
1 vote
Accepted

Approximating an arbitrary set in $\mathbb{R}^n$ with countable unions of cubes?

This becomes essentially a classical result in topology, going under the name normality for topological spaces. Let $d$ denote the metric on $\mathbb R^n$ corresponding to the $\ell_\infty$-norm. Thus,...
Moishe Kohan's user avatar
1 vote
Accepted

Perelman's proof applicability to wireframe sphere

First of all, it is hopeless to understand the result (I am not talking about the proof) without knowing at least some rudimentary topology (at least topological spaces, homeomorphisms, smooth atlas, ...
Moishe Kohan's user avatar
1 vote

Rational point on a pythagorean rectangle

This is an answer to the initial question. It wanted only rational points on the sides of the given rectangle. We may want to switch $a,b$ to search for points on the side of length $b$ (and first on ...
dan_fulea's user avatar
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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 ...
Hrhm's user avatar
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