# Tagged Questions

**1**

vote

**1**answer

220 views

### Possible homotopy-theoretical approach to Gauss-Bonnet

Consider a vector bundle $V\to E\to X$ with fiber $V$, with structure group $G$, and $X$ path-connected. Consider a connection $\nabla$ on $E$. Then for any loop $L$ in $X$, based at $p$, we have a ...

**15**

votes

**1**answer

409 views

### Does the holonomy map define a homomorphism $\pi_k(X)\to\pi_{k-1}(Hol(\nabla))$?

Consider a vector bundle $V\to E\to X$ with fiber $V$, with structure group $G$, and $X$ path-connected. Consider a connection $\nabla$ on $E$. Then for any loop $L$ in $X$, based at $p$, we have a ...

**2**

votes

**1**answer

230 views

### Homotopy versus path-homotopy on punctured surface

I have some problems with homotopies.
The situation is this:
Let $X$ be a surface, which is homeomorphic to a 2-Sphere with a finite number (at least 3) of points removed (equivalently, an open ...

**0**

votes

**2**answers

287 views

### even dimensional manifold homotopic to a symplectic or complex manifold

I have the following question: Let $M$ be an even dimensional Riemannian manifold. Under which conditions does there exists a homotopy to some symplectic manifold? is there any chance that such a ...

**1**

vote

**3**answers

406 views

### prequantization on $TM \bigoplus T^*M$

Let $M$ be a pre-symplectic manifold.In recent years several Geometrists are working on $TM \bigoplus T^*M$ which has fascinated the complex and Poisson geometry. In recent decade also Nigel Hitchin ...

**2**

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**2**answers

265 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 ...

**20**

votes

**4**answers

1k views

### What is an $(\infty,1)$-topos, and why is this a good setting for doing differential geometry?

In this post on the n-Category CafĂ©, Urs Schreiber says that, "The theory of G-principal bundles makes sense in any $(\infty,1)$-topos." I followed the link to the nLab and tried to chase definitions, ...

**4**

votes

**2**answers

332 views

### (Infinite) Suspension Functor on the Pontryagin-Thom Construction

This is a slightly revamped version of a question I asked on the stackexchange forum. That question was asking if the Pontryagin-Thom constructon respects the suspension operation, alluding to stable ...

**6**

votes

**2**answers

370 views

### Proper maps and transversality

I'll begin with the question, which is intrinsically interesting:
Let M be a manifold with some submanifold Y. Suppose that $W \rightarrow M$ is a smooth, proper map. Does there exist another map ...

**15**

votes

**0**answers

311 views

### Lipschitz constant of a homotopy

Let $n>0$, $\mathbb S^n$ be $n$-sphere and $1\in \mathbb S^n$ be its north pole.
A am looking for an example of compact manifold $M$ with a continuous $n$-parameter family off maps $h_x\colon M\to ...

**6**

votes

**2**answers

378 views

### Can a smooth, immersed loop in R^2 become not nullhomotopic by removing a point?

ATT
More precisely, let $\gamma :S^1\rightarrow R^2$ be a smooth immersed loop, the question is whether it is true that there is a point $p\in R^2-\gamma(S^1)$ such that $\gamma$ is not homotopic to ...

**5**

votes

**1**answer

564 views

### Flat Principal Connections and Homotopy Groups?

I've come across a comment in a paper that hints at a relation between the set of (flat) connections for a principal bundle and the homotopy group of the base of the bundle. Can anyone tell me what ...

**13**

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**4**answers

1k views

### homotopy type of connected Lie groups

Is there a simple proof (short and low-tech) of the following fact:
(E. Cartan) A connected real Lie group $G$ is diffeomorphic (as a manifold) to
$K\times\mathbb{R}^n$ where $K$ is a maximal ...

**2**

votes

**2**answers

934 views

### (how) are vector bundles and homotopy groups related?

Hello,
homotopic maps induce isomorphic pullback bundles, and so isomorphism classes of vector bundles over X correspond to homotopy classes of maps from the grassmannian to X. But I think, in the ...