# Tagged Questions

**4**

votes

**3**answers

645 views

### When are maps between topological spaces homotopic?

I wanted to ask if there is any known mehod to quantify 'how many' homotopy classes of maps there are between two given topological spaces $X$, $Y$ (CW-complexes, say).
So far I had the following ...

**5**

votes

**1**answer

199 views

### Is the homeomorphism class of a connected open set of C determined by its fundamental group?

Let $U,U'\subseteq\mathbf{C}$ be two connected open sets such that $\pi_1(U)\simeq\pi_1(U')$.
Q: Does this imply that $U$ is homeomorphic to $U'$?
In the case where the $\pi_1$'s are trivial then ...

**5**

votes

**1**answer

265 views

### Does there exist a space X whose suspension is homotopy equivalent to [0,1] rel ends but where X is not contractible?

As pointed out by David White in
when mapping cone is contractible
there exists an acyclic CW-complex $X$ which is not contractible but whose suspension is contractible. Namely, let $a$ and $b$ be ...

**2**

votes

**1**answer

280 views

### Contractibility of a configuration space

For a topological space $X$ and a positive integer $k\in \mathbb{N}_{>0}$ let $F_k(X):= \{ (x_1,\ldots,x_k)\in X^k |x_i\neq x_j \text{ for } i\neq j \}$ be its $k$-configuration space.
Let ...

**0**

votes

**1**answer

163 views

### Are period domains ever contractible

Which simply-connected period domains are contractible?
Examples. Siegel upper-half space? Poincare upper-half plane? Universal cover of a Shimura variety?
Are these contractible?

**2**

votes

**1**answer

235 views

### Continuous functions on path-connected subsets

Let $X$ be a topological space, and $PX$ the space of all paths on $X$. Then let $G\subset X$ be a path-connected subset and $p\in G$ a point. Let $\sigma:G\rightarrow PX$ be a continuous function ...

**2**

votes

**2**answers

245 views

### Homotopy Equivalences and Induced Correspondences between Fibre Bundles

Suppose that $f:X\rightarrow Y$ is a homotopy equivalence of manifolds. Given a manifold $F$, the pullback construction for $f$ yields a correspondence between isomorphism classes of fibre bundles ...

**4**

votes

**2**answers

310 views

### How to specify a finite group up to inner automorphism?

I want some finite set of data to which I can canoically associate a "group up to inner automorphism", and which can be constructed canoically from a "group up to inner automorphism". I have a few ...

**2**

votes

**1**answer

242 views

### Homotopy groups of K3

Let X be a K3 surface and $Y=X/\mathbb{Z}_2$, an Enrique surface.
Long exact sequence of homotopy groups corresponding to fiberaion $\pi:X\to Y$, says that $\pi_2(X)=\pi_2(Y)$, while we know $H_2(X)$ ...

**27**

votes

**9**answers

2k views

### Covering maps in real life that can be demonstrated to students

Edit: I've really enjoyed everyone's examples (especially the pictures!), but I was mostly looking for a general theorem. For instance, a similar statement to mine is, Can the mapping cylinder of ...

**4**

votes

**1**answer

290 views

### (Non)-exoticness of a diffeomorphism of a sphere

Suppose you have a standard sphere $S^n$ and a "standard" $S^{n-2}\subset S^n$. I am really thinking about $S^{n}\subset \mathbb{R}^{n+1}$ the usual sphere, and $S^{n-2}=S^n\cap \{x_0=x_1=0\}$. Let ...

**3**

votes

**0**answers

183 views

### A proof of the gluing axiom of a TQFT

I posted the following question on math stackexchange but I have not received any answer.
So I hope people here can help me.
In the book, Lectures on tensor categories and modular functors by Bakalov ...

**24**

votes

**2**answers

889 views

### Euler characteristic and universal cover

Let $M$ be a compact manifold, let $\tilde{M}$ be its universal cover, and suppose that the Euler characteristic $\chi(\tilde{M})=0$.
My question is: does this imply that $\chi(M)=0$?
This is clear if ...

**4**

votes

**2**answers

454 views

### Intersection forms of 4-manifolds with boundary

Let $X$ be any simply connected smooth 4-manifold with a fixed Euler characteristic $e$, signature $\sigma$ and boundary $Y$. Assume that the determinant of the intersection form $Q_{X}$ is equal to a ...

**3**

votes

**1**answer

428 views

### The number of simply connected 4-dimension manifold

For a simply connected four-dimension manifold, we know the Freedmen's work.
My question is: For every integer N, Is the number of simply connected 4-manifolds which the second betti number is ...

**1**

vote

**1**answer

304 views

### Homology and homotopy of a surface

Suppose $S$ be a closed orientable genous $g$ surface. Let $f$,$g$ be homeomorphis from $S$ to itself. Assume they induce the same map on 1st homology $H_1(S, \mathbb Z).$
My question is; does this ...

**5**

votes

**2**answers

885 views

### Construction of Serre Spectral Sequence

I'm trying to follow Hopkins' construction of the Serre Spectral Sequence, but some "obvious" things are not that obvious to me.
He starts with considering a double complex $C_{\bullet,\bullet}$ with ...

**12**

votes

**3**answers

454 views

### Recognizing the 4-sphere and the Adjan--Rabin theorem

The problem of recognizing the standard $S^n$ is the following:
Given some simplicial complex $M$ with rational vertices representing a closed manifold,
can one decide (in finite time) if $M$ is ...

**5**

votes

**0**answers

524 views

### Homotopy groups of a Bouquet of n-spheres

Let $$X=\vee_{\alpha\in A} S_{\alpha}^n$$ be a bouquet of $n$-spheres.
Q: How does one compute the homotopy groups $\pi_k(X)$?

**-1**

votes

**1**answer

420 views

### Is this manifold orientable? [closed]

Let $C$ be the set of points $(a,b,c,d) \in \mathbb{C}^4$ which satisfy
1) $ \left|a\right|^2+\left|c\right|^2=\left|b\right|^2+\left|d\right|^2 =1 $.
2) $ a\bar{b}+c\bar{d}=0 $
There is a ...

**0**

votes

**1**answer

312 views

### A form of Lefschetz duality

Let W be a manifold with boundary such that \partial W is a union of two compact manifold A,B attached along their boundary. Does poincare duality hold for (W,A) and (W,B)?

**2**

votes

**0**answers

271 views

### Bipartite Obstructions to genus $g+1$ Bipartite graph from becoming genus $g$.

Say $B_{n,n}$ is a bipartite graph on $2n$ vertices with each color assigned to $n$ vertices.
Say I know $g \le genus(B_{n,n}) \le g+1$: What obstructions can $B_{n,n}$ have from becoming a genus ...

**0**

votes

**1**answer

309 views

### Triviality of finite fiber bundles [closed]

Hello,
I suspect the following is true and easy but I am unable to prove. Suppose (E, B, π, F) is a fiber bundle, where E,B are compact and F is finite, prove that E is a trivial fiber bundle. Any ...

**3**

votes

**1**answer

135 views

### Examples of H-cogroups and a question about julia sets

The following questions should be very easy, but I need help. Notations are same as Bredon's geometry and topology book. This question is related to chapter VII.3.page 441.
$\nabla \colon X\vee ...

**12**

votes

**5**answers

1k views

### Is every real n-manifold isomorphic to a quotient of $\mathbb{R}^n$?

I'm curious about the following:
Is every real n-manifold isomorphic to a quotient of $\mathbb{R}^n$?
Thanks.
EDIT: As Tilman points out, the manifold should be connected. Also, yes, I'm thinking ...