Skip to main content

All Questions

Filter by
Sorted by
Tagged with
23 votes
1 answer
2k views

Is the normal bundle of a torus trivial?

Question: Let $T^k \subseteq \mathbb{R}^n$, $ n > k$, be a smoothly embedded $k$-torus. Is its normal bundle trivial? What about the normal bundle of $S^k \subseteq \mathbb{R}^n$, $n > k$, the $...
Matthew Kvalheim's user avatar
8 votes
2 answers
2k views

Any 3-manifold can be realized as the boundary of a 4-manifold

We know "Any closed, oriented $3$-manifold $M$ is the boundary of some oriented $4$-manifold $B$." See this post: Elegant proof that any closed, oriented 3-manifold is the boundary of some ...
wonderich's user avatar
  • 10.5k
30 votes
2 answers
2k views

Does there exist any non-contractible manifold with fixed point property?

Does there exist any non-trivial space (i.e not deformation retract onto a point) in $\mathbb R^n$ such that any continuous map from the space onto itself has a fixed point. I highly suspect that the ...
Anubhav Mukherjee's user avatar
12 votes
0 answers
460 views

3 manifolds with diffeomorphic unit tangent bundles

What can one say about two closed oriented 3-manifolds $M_1$ and $M_2$ such that $S^2 \times M_1$ is diffeomorphic to $S^2 \times M_2$?
Murat Saglam's user avatar
2 votes
1 answer
94 views

Density of functions into the circle glueing

Let $\{U_i\}_{i=1}^2$ be an open cover of $S^1$, with $U_i\cong \mathbb{R}$ (for example, $U_1$ is the lower arc of the circle and $U_2$ is the upper part). Let $\iota_i:U_i\hookrightarrow S^1$ be ...
ABIM's user avatar
  • 5,405
2 votes
1 answer
308 views

Does any smooth orbifold can be triangulated by orbi-simplex(triangulation of orbifolds)

every smooth manifold can be triangulated, is it true for orbifold? Is it a known result? If yes, is there any reference? reply to the comment : G does not need to be any subgroup of Sn , any ...
haoyu's user avatar
  • 21