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 ...

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1answer
46 views

Prove $f\colon X/{\sim} \to Y \text{ is continuous} \iff \pi\circ f\colon X \to Y \text{ is continuous}$

I need to show that $$f\colon X/{\sim} \to Y \text{ is continuous} \iff \pi\circ f\colon X \to Y \text{ is continuous}$$ where $X/{\sim}$ is a quotient topology and $\pi$ is the quotient map. I ...
1
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1answer
71 views

How can you prove that the winding number around two zeros of a vector field is the sum of the two indices?

If v is a continuous vector field with two isolated zeros, then the winding number around one zero is its index. The winding number on a circle with both zeros in its interior is the sum of the two ...
14
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2answers
476 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. ...
4
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0answers
355 views

The Birman–Hilden Theorem and the Nielsen–Thurston classification

So this post is half question/half reference request, as I'm sure it's the kind of thing people would have thought about before (and indeed the question might even be trivial), but I've been unable to ...
5
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3answers
132 views

when is the region bounded by a Jordan curve “skinny”?

How can I formalize and prove the following intuition?: Picture a very skinny rectangle, one with base length 1 and sides length $\epsilon$. Or imagine a very flattened ellipse. The interiors of ...
2
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1answer
80 views

Density of continuous knots in the plane transversal to some circles

This is an exercise from the book "Knots and Links" by Rolfsen (exercise 6 in section 2C) Let $\kappa : S^1 \rightarrow \mathbb{R}^2-(0,0)$ be a continuous imbedding. Let $M := \{ x \in ...
8
votes
1answer
160 views

Equivalence of Definitions of Principal $G$-bundle

I've finally gotten around to learning about principal $G$-bundles. In the literature, I've encountered (more than) four different definitions. Since I'm still a beginner, it's unclear to me whether ...
4
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0answers
229 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 ...
2
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1answer
71 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) ...
3
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1answer
146 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 ...
2
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0answers
73 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 ...
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0answers
53 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 ...
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1answer
162 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
3
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1answer
192 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 $ ...
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0answers
55 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 ...
4
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1answer
149 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
60 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
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1answer
160 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
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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 ...
5
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1answer
89 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
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1answer
109 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 $\mathbb{Z}/a$?
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1answer
163 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
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1answer
149 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. $2)$ ...
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1answer
130 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
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2answers
233 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 ...
6
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1answer
446 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 ...
2
votes
2answers
249 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 ...
11
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1answer
257 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 ...
2
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0answers
152 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
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1answer
149 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 ...
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1answer
131 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 ...
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1answer
215 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
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1answer
147 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}: ...
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 ...
2
votes
1answer
85 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
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2answers
113 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 ...
2
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0answers
98 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 ...
1
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1answer
104 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 ...
2
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0answers
105 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
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 $S_2$ ...
4
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0answers
110 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 $2-k$. ...
9
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1answer
223 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 ...
5
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0answers
119 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 $\ker(\rho)$ ...
35
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0answers
2k 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
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0answers
131 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 ...
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0answers
40 views

Functionals defined on curves

I am looking for classical (real) functionals defined on rectifiable curves (neither necessarily simple nor closed) in either $\mathbb{R}^2$ or $\mathbb{R}^3$. There's length, turning number, total ...
2
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0answers
68 views

4D TQFT construction from a modular tensor category

I know the construction of 3D topological quantum field theory (TQFT) from a modular tensor category. I heard that we can even (mathematically) construct 4D TQFT from a modular tensor category. I ...
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3answers
287 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 ...
2
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0answers
83 views

Relationship between the Hopf Fibration and Spinors on $S^2$

The unique spin structre for $TS^2$ is given by the Hopf fibration. We can trivialize the Hopf fibration over open sets $U_1 = S^2 \setminus \{N\}, U_2 = S^2 \setminus \{S\}$ where $N$ and $S$ are the ...
2
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1answer
41 views

Homologous tori in 4-manifold

Let $X$ be a 4-manifold and $T$, $S$ two tori embedded in $X$. Let $m_1$, $l_1$ and $m_2$, $l_2$ be loops in $X$ generating $H_1$ of $T$ and $S$, respectively (where I am identifying the tori with ...