4
votes
1answer
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 ...
24
votes
2answers
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
2answers
455 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 ...
1
vote
1answer
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 ...
12
votes
3answers
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 ...
4
votes
3answers
423 views

Flat regions on surfaces of genus greater than 1

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 ...