Tagged Questions
4
votes
1answer
89 views
Representation of (co)homology classes of $3$-manifolds by embedded surfaces
Let $M$ be a closed oriented $3$-manifold. Theorems in algebraic topology allow us to identify
$$H_2(M) \ \cong \ H^1(M) \ \cong \ \langle M,S^1\rangle$$
where (co)homology is meant with integer ...
12
votes
1answer
268 views
A simply-connected closed surface is a sphere
From the Classification Theorem for closed (i.e. compact and boundaryless) surfaces, it follows that $S^2$ is the only closed surface with trivial $\pi _1$. That's easy because the fundamental group ...
3
votes
0answers
59 views
Do we really need to use the Jordan-Schönflies Theorem to prove that every surface can be triangulated?
I have read that most proofs of the triangulability of surfaces require the use of the Jordan-Schönflies Theorem. However, is such high-tech machinery really needed? The problem is that 3-manifolds ...
1
vote
1answer
99 views
Pinched torus generalization
The pinched torus is homeomorphic to a sphere with two (different) points identified.
What is the name and topological structure of the ...
