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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97 views

Manifold geometry and Non - Euclidean geometry

What is the difference between Manifold geometry and Non-Euclidean geometry; what connection is there between them?
6
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1answer
212 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? ...
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1answer
112 views

Compact surfaces and Fundamental Groups

Identify the compact surfaces $X$ for which there exist a proper subgroup $G$ of $\pi_1(X)$ such that $G\cong \pi_1(X)$. EDIT: Suggestions?
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0answers
146 views

Covering spaces!

If one wanted to find all connected covering spaces of a product of two spaces, say $S^1\times RP(3)$, how would you go about it? I'm thinking finding the fundamental group of $S^1\times RP(3)$, and ...
3
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0answers
76 views

Curvature on topological spaces

On what subsets of the category of topological spaces are different notions of curvature defined?
5
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1answer
391 views

Spin structures on $S^1$ and Spin cobordism

I'm trying to understand the 2 spin structures on the circle. Since the frame bundle for the circle is just the circle itself, Spin structures on $S^1$ correspond to double covers of $S^1$. There are ...
3
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0answers
35 views

What is the “real osculating space” of an immersion?

In a differential geometry paper from 1979 I have come across some terminology which I have not found explained anywhere else. We have an immersion $x : S^2 \to S^n$. In the paper, it is a minimal ...
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3answers
141 views

Finding the rank of subgroups of free groups?

How do you find the rank of a subgroup(of finite index) of a free group? I was thinking of looking at the fundamental group of a graph.
2
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0answers
59 views

About Thom theorem (representation submanifold for $H_{n-2}(M)$)

Recall Thom theorem : If $M^n$ is a smooth orientable closed manifold then any homology class in $H_{n-2}(M)$ is represented by the fundamental class of a smooth submanifold. And in the Harper and ...
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1answer
221 views

Higher homotopy groups!

How would you show that $\pi_n, n>1$ of the Klein bottle is the trivial group? I was thinking Seifert-Van Kampen could be applicable?
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87 views

How difficult is it to impose a differential structure on a fractal?

Assuming we have a 4-dimensional smooth manifold $M$ embedded in $\mathbb{R}^{m}$ that is difficult to understand its differential structure. For convenience's sake we can assume it is compact, simply ...
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2answers
178 views

How to obtain a Morse function on a submanifold of Euclidean space

Consider a smooth $n$-dimensional submanifold $A$ in $\mathbb{R}^{n+1} \times \mathbb{R}$ and the projection $f:\mathbb{R}^{n+1} \times \mathbb{R}\rightarrow \mathbb{R}$ onto the second factor. Is it ...
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2answers
153 views

intersection of decreasing path-connected spaces

If we have path-connected spaces $A_0 \supseteq A_1 \supseteq A_2 \supseteq \ldots$, is $\bigcap^\infty A_i$ path-connected? I was thinking that if we take $A_i$ to be a $1/i$-neighborhood of the ...
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0answers
106 views

Generate coordinates for abstract triangulation

I have an abstract triangulation, which consists of nodes without coordinates and connectivity information (the triangles themselves). I also know that each link has a fixed length. For simplicity we ...
3
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0answers
181 views

Fundamental Group!

Say you have two surfaces of genus 2, say $X$ and $Y$ and you want to attach them via homotopy attaching maps $f$ along their waist curves. Then what will the fundamental group of the adjunction space ...
4
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2answers
177 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
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0answers
70 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 ...
4
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1answer
146 views

Prime decomposition of 3-manifolds

Let $H_g$ be a three dimensional handlebody bounded by a genus $g$ surface. Let $M_g$ be a manifold obtained by gluing two copies of $H_g$ via an orientation reversing homeomorphism of the surface of ...
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0answers
233 views

When are maps between topological manifolds automatically surjective?

Take an injective (continuous) map $T:\mathbb{S}_1\to\mathbb{S}_1$. It's an obvious fact (though I can only prove it with nontrivial facts about homotopy) that $T$ is automatically surjective. I have ...
4
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1answer
88 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 ...
7
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3answers
528 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 ...
3
votes
2answers
238 views

line equidistant from two sets in the plane

Suppose $A$ and $B$ are disjoint subsets of the plane, both closed, nonempty, and connected. Define $E(A, B)$ as the set of points in the plane equidistant from $A$ and $B$. For example, if $A$ is a ...
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0answers
52 views

ENR spaces which group of homeomorphisms is not locally contractable

Siebenmann on page 1 of his manuscript gives an example of an ENR space (euclidean neighborhood retract) whose group of homeomorphisms is not locally contactable. Take the sphere ...
3
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2answers
485 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: ...
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2answers
2k views

Retraction of the Moebius strip to its boundary

Prove that there is no retraction (i.e. continuous function constant on the codomain) $r: M \rightarrow S^1 = \partial M$ where $M$ is the Moebius strip. I've tried to find a contradiction using ...
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1answer
54 views

Nonhomeomorphic CW-complexes that are stably homeomorphic

Can one find two finite CW-complexes $X$ and $Y$ such that $X \times I$ is homeomorphic to $Y \times I$, where $I = [0, 1]$, but $X$ is not homeomorphic to $Y$? I know how to find such topological ...
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1answer
448 views

The Join of Two Copies of $S^1$

So I know the fact that the join of $S^1$ and $S^1$ is homeomorphic to the 3-sphere, but I'm having trouble "seeing" this. I'd prefer something that appeals to geometric intuition, but more formal ...
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1answer
94 views

a question about connected sum

I know this probably has a really straightforward answer, specially if it is as standard as it is entuitive to visualize. Still, because i'm not experienced at all on working with this objects and im ...
2
votes
1answer
119 views

Could a surface bundle over a circle have free fundamental group?

Specifically, I was wondering if the surface was non-compact with infinitely generated free fundamental group, could the surface bundle itself have infinitely generated free fundamental group. In this ...
10
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2answers
393 views

$\{(x,y)\!\in\!\mathbb{B}^n; -\varepsilon\leq-\|x\|^2\!+\!\|y\|^2\leq\varepsilon\}\approx\mathbb{B}^k\!\times\!\mathbb{B}^{n-k}$

The question is motivated by the notion of handle attachment, Morse theory, critical points of index $k$, Morse lemma, sublevel sets, etc. For $0\!\leq\!k\!\leq\!n$ and ...
4
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0answers
217 views

Partitioning a triangulated 2-sphere into two triangulated discs

Take a triangulation of the 2-sphere, $S^2$. Let the triangulation be denoted $T$. The Euler characteristic tells you that the number of triangles in $T$ is even. Since triangulations of the ...
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vote
1answer
63 views

Why can we consider only torus switch rather than arbitrary Dehn filling in Lickorish theorem?

I understand that any orientable $3$-manifold can be obtained by doing Dehn surgery on $S^3$ along a set of circles sitting in it; but why can we further assume the slop to be $0$, i.e. we can obtain ...
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1answer
171 views

Given lattice G; parameters of torus R^2/G?

This should be a simple, known result, but I can't seem to find it. Given a lattice $\Gamma = m\mathbb{Z} \times n\mathbb{Z}$, $\mathbb{R}^2/\Gamma$ is topologically a torus. For suitable $m$ and ...
5
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2answers
133 views

Are bounded open regions in $\mathbb{R}^n$ determined by their boundary?

Let $U$ and $V$ be two bounded open regions in $\mathbb{R}^n$, and let us further assume that their topological boundaries are nice enough that they are homeomorphic to finite simplicial complexes. ...
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0answers
45 views

Spinning construction

The p-spinning construction works as follows: You start with $A$ an $n$-ball embedded in an $m$-ball $B$ such that $\partial A \subset \partial B$ and $A^\circ \subset B^\circ$. The $p$-spin of ...
2
votes
1answer
118 views

Morton Brown's theorem

Suppose that a topological space $X$ is the union of an increasing sequence of open subsets $U_i$c each of which is homeomorphic to the Euclidean space $\mathbb{R}^n$. How does one show that $X$ is ...
4
votes
1answer
103 views

Isotopy to the identity on disk

Let $D^2 \subset \mathbb{R}^2$ the unit disk and $f: D^2 \rightarrow D^2$ a homeomorphism with the property that $f$ restricted to the boundary $\partial D^2$ is the identity. Then $f$ is ambient ...
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1answer
61 views

A knot which intersects $S^2$ transversely once in 3-connected manifold

I am reading the paper,ON ATTACHING 3-HANDLES TO A 1-CONNECTED 4-MANIFOLD by BRUCE TRACE here.He says in this paper that we need only construct a knot $K\subset \partial W^4$ which meets $Σ ^2$ ...
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0answers
97 views

About manifolds after attaching handles.

I'm reading R.E. Gompf and A.I. Stipsicz, 4-Manifolds and Kirby Calculus. I don't understand Remarks 4.4.1 on page 116-117 Google books here. At first I can't understand why we take immersed disk $D ...
4
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1answer
150 views

About Kirby Diagrams

I'm reading R.E. Gompf and A.I. Stipsicz, 4-Manifolds and Kirby Calculus. There is something I don't understand on page 116 (Google Books link to page 116; alternatively, here are images of page 115 ...
13
votes
1answer
380 views

Can one cancel $\mathbb R$ in a bi-Lipschitz embedding?

Let $X$ be a metric space. Suppose that the product $X\times\mathbb R$ admits a bi-Lipschitz embedding into $\mathbb R^{n}$. Does it follow that $X$ admits a bi-Lipschitz embedding into $\mathbb ...
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0answers
190 views

Hairy ball theorem: references to applications

I'm looking for references to applications of the Hairy ball theorem. I already visited wikipedia and cited references, but I need a little more explanation in both meteorology and applications in ...
6
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1answer
236 views

Question on “up to isotopy” when attaching two spaces

Let $M$, $N$, $A$, $B$ be topological spaces (or manifold) such that $A$ and $B$ are subspaces in $M$, $N$ respectively. Let $f: A \to B$ and $g:A \to B$ be homeomorphism and assume that $f$ and $g$ ...
11
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1answer
412 views

Covering points on a sphere with a disk

Suppose $m$ points ("sites") are selected on the unit sphere $S^2$. For a given radius $r < \pi$, we can define a disk around any point on the sphere as the set of points at geodesic distance at ...
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0answers
112 views

Multiple Dehn twists and minimal position

I have a question about a proof that I am reading in "A primer on Mapping Class Groups" by Farb and Margalit. Let $a$ be a simple closed curve in a compact surface $S$ (possibly with marked points ...
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1answer
113 views

Bigon question related to Dehn twists

Perhaps someone can help me with this: For simple closed curves on an orientable compact surface, if they form a bigon, then is it true that at the intersection points the orientations must be ...
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1answer
557 views

For an $n$-dimensional object, how many types of holes are possible?

Update 2012-06-06: At some point I'll attempt to answer my own question by using a dual-fluid model that places the dimensionality and connectivity of "solids" and "holes" on an equal footing. With ...
2
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1answer
127 views

Poincaré conjectures for other shapes

If you replace spheres in the Poincaré conjecture with objects-with-some-kinds-of-holes can you say that every manifold with the same number and type of holes is homeomorphic to every other such ...
3
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1answer
179 views

Topological shape of sphere-like equations with complex radius

For fixed complex number $s≠0$, what 4-dim shape is given by complex solutions $z,w$ of $z^2+w^2=s^2$ and higher dimensional version 2N-dim shapes of $z_1^2+z_2^2+...+z_N^2=s^2$ ?
11
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1answer
557 views

Why is the knot group of the trefoil isomorphic to the group of 3-braids?

I apologise in advance for the vagueness of this question but I have not been able to find very much info on the topic and have made very little progress on my own. I am trying to understand why the ...