# All Questions

Tagged with gt.geometric-topology homotopy-theory

79 questions

**1**

vote

**0**answers

213 views

### On the homotopy type of $\mathrm{Diff}(\mathbb{S}^3)$

I am confused with the following argument. I know I am doing something wrong but I can't find my mistake.
On one hand, one knows that if $M$ is a Lie group, then
$$\mathrm{Diff}(M)\simeq M\times\...

**2**

votes

**0**answers

88 views

### Space of non-vanshing sections path-connected?

Let $M$ be a path connected smooth manifold and $E$ be a vectorbundle over $M$ of rank at least two. My question is: Under which conditions is the space of global non-vanishing sections path connected?...

**6**

votes

**0**answers

243 views

### Quotient space, a fundamental group, and higher homotopy groups 2

Previously, I ask for comments/suggestions on setting up the calculation in Quotient space, homogeneous space, and higher homotopy groups. There, however, I was looking for whatever methods and tools ...

**2**

votes

**0**answers

254 views

### Quotient space, homogeneous space, and higher homotopy groups

Preparation and my input:
For the quotient space $G/H$, knowing the homotopy
groups of $G$ and $H$ one can determine homotopy groups from the long
exact sequence
$$
...
\to \pi_n(H) \to \pi_n(G) ...

**5**

votes

**0**answers

176 views

### Asphericity of hypersurface complement in ${\mathbb C}^n$

How does one check that the following space is aspherical?
$X_n=\{(x_1,x_2,\ldots , x_n)\in {(\mathbb C^*)}^n\ |\ x_i\neq x_j\ and\ x_ix_j\neq 1\ for\ i\neq j\}$.
One way I can think of is to give ...

**5**

votes

**0**answers

65 views

### Bounding the dimension of the euclidean space in which any $n$-manifold embeds “$k$-uniquely” in it

(The question will be interesting for topological/Pl as well but in order to not be too vague I will restrict the meaning of manifold to smooth manifold without boundary).
I'm interested in the ...

**2**

votes

**1**answer

255 views

### Are there compact flat fiber bundles with “truly” infinite structure group?

Let $p: E \rightarrow B$ be a flat fiber bundle with fiber $F$ where $E$, $B$, $F$ are compact, smooth manifolds.
I am looking for a counterexample for the following statement:
$E \cong \widetilde{B}...

**5**

votes

**0**answers

252 views

### What is the local structure of a fibration?

It's sometimes said that a fibration is a fiber bundle which is not locally trivial. I'd like to make this precise, by identifying the "local models" on which fibrations are modeled.
Here I'd like ...

**19**

votes

**2**answers

1k views

### A manifold is a homotopy type and _what_ extra structure?

Motivation: Surfaces
Closed oriented 2-manifolds (surfaces) are "classified by their homotopy type". By this we mean that two closed oriented surfaces are diffeomorphic iff they're homotopy ...

**3**

votes

**0**answers

106 views

### Weak contractibility of some infinite dimensional metric spaces

Let $(X_{n},d_{n})_{n \in \mathbb{N}}$ be a sequence of complete geodesic metric spaces such that:
$X_{n}$ is a regular$^1$ CW-complex of constant local dimension$^3$ $n$, it is of finite type$^4$, ...

**9**

votes

**4**answers

601 views

### Complements of Simply Connected Subsets of the Plane

this is my first question here! Hopefully it is appropriate. Let $\mathbb{A}$ be the punctured plane, i.e. the 'standard' annulus. For compact, connected subsets of the plane (planar continua) $X \...

**21**

votes

**2**answers

422 views

### Morphism from a surface group to a symmetric group, lifted to the braid group

Let $\Sigma_g$ be the fundamental group of the closed orientable surface of genus $g\ge 2$; let $B_n$ be the braid group on $n\ge 3$ braids; let $S_n$ be the symmetric group on $n$ letters; let $p:B_n\...

**2**

votes

**1**answer

133 views

### Classification of pretzel links up to link homotopy using alexander quandle

I am currently reading this paper where the author classifies the pretzel links up to link homotopy using a quasi-trivial quandle $\mathbb{Z}_{k}[t^{\pm 1}]\diagup_{(t-1)^{2}}$, and I find it ...

**3**

votes

**1**answer

255 views

### Linking number a complete invariant of link homotopy

I read in Milnor's article "Link groups", where he defines invariants to classify links up to link homotopy, that the linking number is a complete invariant which can tell almost trivial two ...

**7**

votes

**1**answer

390 views

### Isomorphism from $\mathbb{Z}$ to third homotopy group of compact simple Lie group

Let $G$ be a compact connected simple Lie group. It is known that its third homotopy group $\pi_3(G)$ is isomorphic to $\mathbb{Z}$. More precisely, there is a Lie group homomorphism
$$\rho:SU(2)\...