The corpus of tools and results that arose from studying manifold theory using non-algebraic techniques, that is, as opposed to (algebraic-topology). The focus of the field tends to be on special objects/manifolds/complexes and the topological characterisation and classification thereof. A key ...
4
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0answers
60 views
Soft question: why are there non-smooth manifolds?
Topologists are often very good at explaining the geometric intuition behind certain results and programs of research. For instance, the particular interest in 4 manifolds is often explained by ...
49
votes
1answer
2k views
Trigonometric sums related to the Verlinde formula
Original question (see also the revised, possibly simpler, version below): Let $g > 1, r > 1$ be integers. Playing around with the Verlinde formula (see below), I came across the expression
...
2
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1answer
32 views
Uniqueness of “Punctured” Tubular Neighborhoods (?)
Here is a question that has been haunting me for a while: Let $\mathbb{R}^{n-1} \times [0, \infty)$ be the upper half space of $\mathbb{R}^n$ and suppose we have a smooth homeomorphism (not a diffeo) ...
2
votes
1answer
50 views
Annulus Theorem
I'm trying to read Rolfsen's "Knots and Links" and I'm a little discouraged that I can't do one of the first and seemingly more important exercises. The question is
Use the Schoenflies theorem ...
8
votes
1answer
171 views
Nontrivial h-cobordism
I'm learning the h-cobordism theorem as I want to use it in a talk. I'd like to be able to give an example of an h-cobordism that isn't a cylinder, if possible by drawing a picture. What is the ...
12
votes
1answer
267 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
240 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 ...
5
votes
1answer
99 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 ...
1
vote
1answer
140 views
Infinitely sheeted covering spaces!
I was wondering what the fundamental group of an infinitely sheeted covering space of say some surface might be?
I'm thinking it should be an infinite cyclic group, but this is more intuitively, i ...
1
vote
0answers
24 views
seek visual pictures or video on decomposition of manifolds
In my study of knot theory, I notice that I lack examples to show some classical decomposition theorems in 3-dimensional manifolds, such as JSJ decomposition theorem, Milnor's prime decomposition ...
10
votes
4answers
755 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 ...
6
votes
1answer
127 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?
...
2
votes
0answers
22 views
$\epsilon$-net of a $n$-dimensional $\ell_2$-ball
Let $B$ be an $\ell_2$-ball of radius $r$ in $\mathbb{R}^n$.
I want to find the cardinal of a (not too big) $\epsilon$-net of $B$, that is the cardinal of a finite set $V\subset B$ such that $\forall ...
0
votes
0answers
31 views
Finding a closed curve $\gamma$ on a Torus such that $\gamma$ is not pseudo-Anosov
Let g denote genus and n denote number of punctures. According to Kra's construction in Pseudo-Anosov theory for surfaces, if S is an orientable surface such that $3g+n>3$ and $\gamma \in ...
1
vote
1answer
85 views
Is there an example of a non-orientable group manifold?
Basically what I'm looking for is a topological group that is also a non-orientable, n-dimensional manifold
2
votes
0answers
46 views
3-manifolds fibering over the circle and mapping tori
If $S$ is a closed connected surface and $\varphi \in \mathrm{Diff}(M)$, then we can build the mapping torus $M_\varphi = \dfrac{S \times [0,1]}{(x,0)\sim (\varphi(x),1)}$. Then we have that $ ...
0
votes
0answers
29 views
PL and triangulizable
Is it correct that the notion of triangulizable manifold (in the sence "homeomorphic to a simplicial complex") is weaker than the notion of a PL-manifold?
If yes, why? (eg is it true that a star of ...
3
votes
1answer
54 views
What is the proof that SO(2n+1) is non-orientable for any positive integer n?
Inquiring minds want to know. :) I know for certain that SO(3) is not orientable but I did read somewhere that for any odd dimension N>1, SO(N) is a non-orientable manifold. If such is true I'm eager ...
2
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0answers
36 views
How do we check if a covering of an orbifold is a manifold?
Let $X$ be an orbifold and suppose it is "good", i.e. its universal covering orbifold $\widetilde{X}$ has a trivial orbifold structure (it is "just" a manifold). It may be the case that some ...
1
vote
1answer
47 views
Immersing punctured torus
I am looking for a proof (as elementary as possible) of the fact that punctured n-dimensional torus admits an immersion to $\mathbb{R}^n$. The 2-dimensional case seems to be evident, but I haven't got ...
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 ...
5
votes
1answer
53 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 ...
4
votes
1answer
50 views
Covering spaces of Lens spaces
Let $L(p,q)$ be the Lens space with composite $p$, say $p=ab$.
What is the cyclic covering space of $L(p,q)$ induced from the quotient group homomorphism from $\mathbb{Z}/p$ to $\mathbb{Z}/a$?
3
votes
1answer
65 views
Gluing axiom of a TQFT
In the book, Lectures on tensor categories and modular functors by Bakalov and Kirillov they construct a TQFT.
When they come to prove the gluing axiom, they just mention that "...This statement is ...
0
votes
1answer
62 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 $N \subset M$.
...
11
votes
1answer
56 views
Equivalence of definitions of $S^\infty$
Consider the following two definitions of the infinity-sphere $S^\infty$. Why do they define homeomorphic spaces?
$1)$ The set of points in $\mathbb R^\infty$ with distance $1$ from the origin.
$2)$ ...
1
vote
1answer
57 views
Simple Sphere Suspension Question
I have heard it said that the suspension of an $n$-sphere is an $n+1$-sphere. This is stated without proof in chapter 0 of Hatcher's book on algebraic topology. More generally, it seems that the ...
4
votes
2answers
101 views
How does Thurston's geometrisation conjecture imply Poincaré's conjecture?
I ran into the geometrisation conjecture a few days ago, and I started wondering how to prove Poincaré's conjecture. Let $M$ be a compact, simply connected, $3$-manifold. Clearly it is irreducible ...
10
votes
1answer
177 views
Space of homeomorphisms Homeo$(S^1)$ of $S^1$ deformation retracts onto $O(2)$
How can we prove that the space of homeomorphisms Homeo$(S^1)$ of $S^1$ (strong) deformation retracts onto the orthogonal group $O(2)$?
I know that this result is proved by Hellmuth Kneser in his ...
4
votes
1answer
99 views
Can someone give an example of a non-differentiable manifold?
A topological space $M$ is a manifold of dimension $n\geq 1$ iff it is a second countable space that is locally homeomorphic to the Euclidean space $R^n$. So if $M$ is a manifold there exists a map ...
1
vote
2answers
78 views
Smooth embeddings that are homeomorphisms but not diffeomorphisms
I'm looking for examples of smooth proper embeddings between connected compact manifolds of the same dimension that are not diffeomorphisms. I remember having seen an example with $S^7$ in ...
0
votes
0answers
19 views
vertex linking sphere
S.Choi in his article " Geometric structures on low dimensional manifolds " uses " Haken diagram " of triangulated 3-manifolds.He starts with a tetrahedron in the triangulation and form the linking ...
2
votes
0answers
42 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 ...
5
votes
1answer
97 views
Cancelling 3-handles in Kirby diagrams
Recently been trying to understand the proofs of Gompf and Akbulut that certain 4-manifolds are $S^4$ (these 2 papers: Gompfs paper in Topology Vol. 30 Issue. 1, Akbulut). In which they use a clever 2 ...
1
vote
1answer
69 views
How is PL knot theory related to smooth knot theory?
I really want to like knot theory but the PL condition seems sort of ugly. I was hoping someone could give me a justification for secretly thinking about smooth knots as I read through a book like ...
1
vote
1answer
71 views
how to prove that quotient group R/Z and the circle are diffeomorphism?
Are quotient group R/Z and the circle are diffeomorphism ? How to prove? Hope someone give me some advise or some reference documents.Thank you
0
votes
1answer
60 views
Stereographic projection of ellipsoid
I am really new in geometry and especially in working with stereographic projection, so excuse me, please, if my question is too easy.
Given is the ellipsoid: $E = \left \{ (x,y,z)\in \mathbb{R}^{3}: ...
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 ...
2
votes
1answer
48 views
Degree of continuous mapping via integral
Let $f \in C(S^{n},S^{n})$. If $n=1$ then the degree of $f$ coincides with index of curve $f(S^1)$ with respect to zero (winding number) and may be computed via integral
$$
\deg f = \frac{1}{2\pi ...
2
votes
2answers
106 views
Compact subvarieties in $\mathbb{C}^n$
I ran across a statement, the maximum principle, which states $X\subset \mathbb{C}^n$ is compact in the Euclidean topology iff $X$ is a finite set of point.
Unfortunately, a proof didn't come along ...
3
votes
2answers
223 views
Homotopy Question Help?
Let $X$ be a topological space and suppose $X_1$ and $X_2$ are spaces obtained by attaching an n-cell to $X$ via homotopic attaching maps. Show that $X_1$ and $X_2$ are homotopy equivalent.
Proof:
...
2
votes
0answers
72 views
Why does a circle cut a torus into an annulus?
Let $\phi : S^1 \rightarrow T^2$ be an (topological. Not necessarily smooth) imbedding of the circle in the 2-torus and let $\iota : S^1 \rightarrow T^2, \theta \mapsto (\theta,0)$ be the imbedding ...
0
votes
1answer
53 views
How to proof M is a compact connected n-manifold,if for any point $p\in M,M\backslash\{p\}\cong R^n$ then M is homeomorphic to $S^n$.
M is a compact connected n-manifold,if for any point $p\in M,M\backslash\{p\}\cong R^n$ then M is homeomorphic to $S^n$.
I have the guess from ...
1
vote
0answers
71 views
Homeomorphism between simply connected, closed 3 - manifold and 3-sphere.
The Poincare conjecture states that a simply-connected, closed 3-manifold is homeomorphic to the 3-sphere. Now that the conjecture has been settled, could someone tell me what this homeomorphism is ...
2
votes
1answer
57 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 $S_2$ ...
4
votes
0answers
73 views
Problem from Hempel — surfaces in Lens spaces
This isn't a homework problem, just working through Hempel's 3-Manifolds for my own benefit. One exercise is to show that the lens space $L(2k,q)$ contains a surface of Euler characteristic $2-k$.
...
5
votes
0answers
78 views
$\operatorname{Spin}^c(n)$ is a Lie group?
Assume that $n\geq 3$. (This implies $\pi_1(\operatorname{SO}(n))=\mathbb{Z}/2$.)
Let $\rho\colon \operatorname{Spin}(n)\to \operatorname{SO}(n)$ be the 2-fold universal cover. Identify $\ker(\rho)$ ...
18
votes
0answers
479 views
Grothendieck 's question - any update?
I was reading Barry Mazur's biography and come across this part:
Grothendieck was exceptionally patient with me, for when we first met I knew next to nothing about algebra. In one of his first ...
4
votes
0answers
92 views
Symmetric product of genus 2-surface
Let $\Sigma$ be the genus 2-surface.
Denote $\operatorname{Sym}^2(\Sigma)=\Sigma\times \Sigma/\mathbb{Z}/2$, where $\mathbb{Z}/2$ acts on $\Sigma\times \Sigma$ by $(x,y)\mapsto (y,x)$.
In the very ...
4
votes
0answers
84 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 ...
