2
votes
1answer
121 views

A 3-manifold with fundamental group isomorphic to a surface group.

Let $M$ be a 3-manifold (the case I am interested is $M$ closed orientable connected hyperbolic); suppose $\pi_1 (M)$ is isomorphic to the fundamental group of a (closed orientable connected) surface ...
0
votes
0answers
19 views

For any diagram of a surface-knot, what are the possible connections of edges?

The singular set of a surface-knot diagram is a disjoint union of i) a graph, which has 1- and 6-valent vertices corresponding to branch points and triple points respectively. ii) circles which has no ...
0
votes
0answers
62 views

The Geometrical Mean of $T^2$#$RP^2=RP^2$#$RP^2$#$RP^2$

This is my homework: $T^2$#$RP^2=RP^2$#$RP^2$#$RP^2$ Below is what I have think: From the book of M.A.Armstrong, I find $RP^2$#$RP^2$#$RP^2$ is a sphere which is attached by three Mobius ...
4
votes
1answer
166 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
476 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
1answer
54 views

Compact orientable surfaces embeddings

I wonder if we can embed a compact orientable surface of genus $g$ into another of genus $g'$, if $g < g'$. I already know that this is false if $g>g'$, because of the first homology groups. Any ...
2
votes
1answer
160 views

Supposedly “trivial” implication that injective surfaces are incompressible

My question is about a passage in Algorithmic Topology and Classification of 3-Manifolds by Sergei Matveev. Let $F$ be a surface in some $3$-manifold $M$. $F$ is called incompressible if for every ...