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**3**

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### Existence of tubular neighborohoods of locally flat topological embeddings

Suppose $X$ is a topological manifold and $Y \subset X$ is a locally flat submanifold. We know that $Y$ doesn't necessarily have a tubular neighborhood. My definition of a tubular neighborhood of $Y$ ...

**2**

votes

**1**answer

46 views

### Shrinkable decompositions with uncountably many non-degenerate elements?

Let $\mathcal D$ be an upper semicontinuous decomposition of $\mathbb S^n$ and let $\mathcal D'\subset\mathcal D$ be the set of non-singletons. The decomposition space $^{\mathbb S^n}/_{\mathcal D}$ ...

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**1**answer

256 views

### Piecewise linear (PL) structures on $\mathbf R^4$

One can read in Wikipedia that the 4-dimensional affine space $\mathbf R^4$ has uncountably many piecewise linear structures (in contrast with other dimensions, where it has exactly one). A reference ...

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**0**answers

198 views

### Is the quotient map of the action of homeomorphisms on embeddings well-behaved?

It is well known that if $M$ and $N$ are smooth manifolds, the diffeomorphisms $Diff(M)$ act continuously on the smooth embeddings $Emb^{C^\infty}(M,N)$ by precomposition, if both are given the ...

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votes

**1**answer

210 views

### A homological criterion for collapsibility?

On page 256 of Kirby and Siebenmann one finds the following lemma (its proof an "elementary exercise", so they only give a hint):
Taking $A$ to be a point and iterating this collapsing lemma, this ...

**27**

votes

**2**answers

695 views

### good covers of manifolds

It is well-known and easy to prove (see for instance this post) that every smooth manifold admits a "good cover", i.e. a locally finite cover by open balls such that all nonempty intersections of the ...

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votes

**2**answers

306 views

### The group of diffeomorphisms with compact support

Let $M$ be a topological/differentiable manifold. Is there any topology on the group of homeomorphisms/diffeomorphisms with compact support, turning it into a (locally-)compact topological group?
(My ...

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**1**answer

690 views

### How can gauge theory techniques be useful to study when topological manifolds can be triangulated?

I was reading a review article arXiv:1310.7644 and it was explained there that in the last few years it was proven that there are topological manifolds of dimension greater than four that cannot be ...

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**3**answers

798 views

### Does a *topological* manifold have an exhaustion by compact submanifolds with boundary?

If $M$ is a connected smooth manifold, then it is easy to show that there is a sequence of connected compact smooth submanifolds with boundary $M_1\subseteq M_2\subseteq\cdots$ such that ...

**10**

votes

**1**answer

222 views

### orbit space of a topological manifold

Given a compact Lie group G acting freely on a topological manifold M, is it true that the orbit space M/G is also a topological manifold? If so, why?

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### When is a compact topological 4-manifold a CW complex?

Freedman's $E_8$-manifold is nontriangulable, as proved on page (xvi) of the Akbulut-McCarthy 1990 Princeton Mathematical Notes "Casson's invariant for oriented homology 3-spheres".
Kirby showed that ...