# Tagged Questions

224 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 ...
238 views

### sections of tensor product bundle ( tensor product of two vector bundles ) [closed]

Suppose we have a smooth manifold M and E--->M is a vector bundle. A connection on E is a linear map from the set of all smooth section on E into the set of smooth sections of the tensor product of E ...
228 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 ...
277 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 ...
859 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 ...
226 views

### geodesics in Lens spaces

For which Lens spaces, all the simple closed geodesics have the same length? Is it just $S^n$ and $S^n/\mathbb{Z}_2$ or there are more?
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### What is the usual topology of $C^\infty_c(M)$

If $M$ is a smooth paracompact manifold, then what is the usual topology of $C^\infty_c(M)$, i.e., the smooth function with compact support?
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### degenerating immersion

Hi, I would like to know if it exists a sequence of $C^2$ immersion $f_k: S^2 \rightarrow \mathbb{R}^3$ which converge (in C^2) to $z^2$ except on a finite set of point, i.e $f^k \rightarrow z^2$ in ...
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### Symplectic structures on a homotopy complex projective space

For $n>2$, there are infinitely many differentiable structures on the homotopy type of $\mathbb{C}P^n$. I want to know which differentiable structures support a symplectic form. My question is as ...
885 views

### Are there topological restrictions to the existence of almost quaternionic structures on compact manifolds?

I start with some background, but people familiar with the subject may jump directly to the question. Let $M^{4n}$ be a compact oriented smooth manifold. Recall that an almost hypercomplex structure ...
529 views

### Varieties, Frechet Completions, and Regular Functions

Take an algebraic variety $V$, and its set of smooth functions $C^{\infty}(V)$. One can endow $C^{\infty}(V)$ with a canonical locally convex topology (the seminorms are defined using the local ...
Here is a polygonal disk + gluing scheme model of a surface we are attempting to construct. We want the regions of the surface bounded by the two vertical, dotted lines $\alpha,\beta$ to have zero ...