2
votes
0answers
50 views

Homologous surfaces in three-manifolds

Let M be a 3-manifold. Let $S$ and $T$ be properly embedded surfaces in $M$ such that $[S] = [T] \in H_2(M, N(\partial S)) $. Is it true that we can isotope $\partial S$ so that it coincides with ...
2
votes
0answers
84 views

4-manifold: $0$-handle $\cup$ $2$-handles along a framed link in $S^3$ (intersection form = linking matrix = presentation matrix of $H_1(\partial M)$)

Let $L$ be a framed link in $S^3$, consisting of framed knots $L_1,\ldots,L_m$. Let $A=[a_{ij}]\in\mathbb{Z}^{m\times m}$ be its linking matrix, with $a_{ii}=$ framing of $L_i$ and $a_{ij}=$ linking ...
1
vote
0answers
49 views

Kirby diagrams for nonorientable $4$-manifolds

In http://www.math.msu.edu/~akbulut/papers/akbulut.lec.pdf, which is a (still developed) set of lecture notes on 4-manifolds by Selman Akbulut, in section 1.5 there is a way to draw a non-orientable ...
11
votes
1answer
208 views

The “Easiest” non-smoothable manifold

In 1960, Kervaire found the first example of a PL-manifold which does not admit a smooth structure. Since then, I understand that there are many examples of non-smoothable manifolds that can be built. ...
5
votes
1answer
83 views

Embedding manifolds of constant curvature in manifolds of other curvatures

I know that there is no complete surface embedded in $\mathbb{R}^3$ of constant curvature -$k$ for any $k$. But you can clearly embed the hyperbolic plane (curvature -1) into hyperbolic 3-space ...
0
votes
1answer
128 views

Submanifolds of Orientable Manifolds With Boundary

Let $(M, \partial M)$ be an orientable $n$-dimensional topological manifold with boundary. Suppose that $(N, \partial N)$ is an $n$-dimensional topological manifold with boundary and $N \subset M$. ...
2
votes
0answers
93 views

Combinatorial surfaces and manifolds

Before we can start some basic definitions to come into the topic: Suppose $T$ is the standard closed triangles, the convex hull of the three basic vectors inside $\Bbb{R}^3$. Consider $T$ is the ...
2
votes
1answer
62 views

Properties of $S_2$ and the plane and $[−1,1]^2$

The question: Is the sphere $S_2$ isometric / diffeomorphic / homeomorphic to the plane? Is the sphere $S_2$ minus a point isometric / diffeomorphic / homeomorphic to the plane? Is the sphere $S_2$ ...
6
votes
1answer
177 views

Orientability of Manifolds

Given that $f \colon \mathbb R^n \rightarrow \mathbb R$ is a smooth function and if $c \in \mathbb R$ is a regular value how would I go about showing that $f^{-1} (c)$ is an orientable manifold? ...
4
votes
2answers
159 views

What is smoothness needed for?

We can either define a knot to be (1) a smooth embedding $S^1 \hookrightarrow \mathbb R^3$ or (2) a piecewise linear, simple closed curve in $\mathbb R^3$ Then these two definitions are ...
3
votes
0answers
66 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 ...
7
votes
3answers
339 views

Embedding compact (boundaryless?) n-manifolds in n-dimensional real space

I know the embedding theorems that allow you to embed $n$-manifolds into $\mathbb{R}^k$, provided $k$ is chosen large enough. Here I'm interested in the possibility of taking $k=n$ in the case of ...
5
votes
1answer
117 views

Status of PL topology

I'm starting to learn about geometric topology and manifold theory. I know that there are three big important categories of manifolds: topological, smooth and PL. But I'm seeing that while topological ...
3
votes
1answer
388 views

Showing a bijective, continuous function between connected, locally euclidean spaces is a homeomorphism.

This question comes from Conlon's Differentiable Manifolds (it's Exercise 1.1.13). Let $X$ and $Y$ be connected, locally Euclidean spaces of the same dimension. If $f:X \rightarrow Y$ is bijective ...
10
votes
4answers
991 views

Reference on Geometric Topology

Geometric topology is more motivated by objects it wants to prove theorems about. Geometric topology is very much motivated by low-dimensional phenomena -- and the very notion of low-dimensional ...
5
votes
0answers
106 views

Heegaard Splitting of Brieskorn homology 3-spheres

For pairwise coprime integers $(p,q,r)$, we define a Brieskorn homology 3-spheres by $\Sigma(p,q,r)=\{(z_1,z_2,z_3)\in\mathbb{C}^3 | z_1^p+z_2^q+z_3^r=0, |z_1|^2+|z_2|^2+|z_3|^2=1\}$. I want to know ...
1
vote
0answers
60 views

Fundamental Domain of Manifold Reference Request

I am interested in learning about the fundamental domain of a manifold and I am wondering if anyone know of any papers or descriptions online other than Wikipedia and the linked articles? I am looking ...
2
votes
0answers
66 views

Upper bound on the number of charts needed to cover a topological manifold

If $M^n$ is a compact topological manifold (not necessarily with additional structure), is there an upper bound on the number of charts needed to cover $M$ ? Does this bound depend on the dimension of ...
1
vote
2answers
162 views

When does an embedded $2$-torus bound a solid torus in $3$-manifolds?

This is a simple version of the question asked here. Let $M$ be a compact 3-manifold without boundary, $\mathbb{T}^2$ be the standard 2-torus, and $i:\mathbb{T}^2\to M$ be an embedding. (*) Assume ...
16
votes
1answer
394 views

Decomposition of a manifold

As a kind of aside to this question, where one of the answers assumed that if $S^n=X \times Y$ then we can assume that $X$ and $Y$ are manifolds. If we have a manifold $M$, such that $M$ is ...
5
votes
2answers
419 views

Which mapping tori are Seifert manifolds?

According to Orlik's lecture notes on Seifert manifolds (and the Wikipedia page on Seifert fiber spaces), a mapping torus over a 2-torus is a Seifert manifold if and only if it is the mapping torus of ...