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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62
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0answers
4k 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 ...
53
votes
1answer
3k 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 ...
27
votes
3answers
2k views

Why is the Jordan Curve Theorem not “obvious”?

I am horribly confused about Jordan's Curve Theorem (henceforth JCT). Could you give me some reason why should the validity of this theorem be in doubt? I mean for anyone who trusts the eye theorem is ...
25
votes
0answers
697 views

In $n>5$, topology = algebra

During the study of the surgery theory I faced following sentence: Surgery theory works best for $n > 5$, when "topology = algebra". I don't know what is the meaning of topology=algebra. ...
19
votes
1answer
431 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 ...
17
votes
2answers
653 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. ...
17
votes
1answer
341 views

The kernel of free group map to surface group

$G$ is a surface group of genus $g\geq 2$ (the fundamental group of closed orientable surface of genus g). $F$ is a free group of rank $2g$ with basis $\{x_1,\dots,x_{2g}\}$. $\phi$ is a surjective ...
14
votes
4answers
635 views

Useful fibrations

What are the most useful fibrations that one be familiar with in order to use spectral sequences effectively in algebraic topology? There's at least the four different Hopf fibrations and $S^1\to ...
14
votes
1answer
288 views

Is the torus the union of two connected, simply-connected open sets?

Is the torus the union of two connected, simply-connected open sets? A routine computation with the Mayer-Vietoris sequence shows that if so, then their intersection must have exactly three ...
14
votes
1answer
641 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 ...
14
votes
1answer
414 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 ...
13
votes
1answer
191 views

Intuition for Exotic $\mathbb R^4$'s

Today one of my professors told me that $\mathbb R^4$ admits uncountably many non-diffeomorphic differential structures. When I asked him whether there's an intuitive reason to expect a result like ...
13
votes
1answer
306 views

Maps from $D^n$ to $D^n$ with a single inverse set are open.

Let $D^n$ denote the closed unit ball in $\Bbb R^n$. In multiple sources proving Brown's generalized Schoenflies theorem (including a version in the original paper), the following consequence of ...
12
votes
3answers
246 views

Is every self-homeomorphism homotopic to a diffeomorphism?

Given a smooth manifold $M$, is every homeomorphism $M \to M$ homotopic to a diffeomorphism? Hirsch's "Differential Topology" has a proof that every $C^1$ diffeomorphism of smooth manifolds is ...
12
votes
1answer
603 views

Gluing a solid torus to a solid torus with annulus inside.

I was thinking the fact that if two genus $1$ handlebodies (solid tori) are glued via an orientation preserving homeomorphism of boundaries, the resulting manifold depends only on (up to isotopy) ...
12
votes
1answer
133 views

How equivalent are the homeomorphism and diffeomorphism groups of 3-manifolds?

Every topological 3-manifold carries a smooth structure, unique up to diffeomorphism. In addition, "up to homotopy", the maps between them are the same: every continuous map is homotopic to a smooth ...
11
votes
4answers
1k 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 ...
11
votes
1answer
283 views

A quadratic reciprocity formula

Inspired by a problem of calculating explicitly the invariants by Reshetikhin and Turaev for certain 3-manifolds, I have come across the following problem involving Gauss sums: I would like to prove ...
11
votes
1answer
307 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 ...
11
votes
1answer
158 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. $2)$ ...
11
votes
1answer
145 views

Is the E8 manifold homeomorphic to a CW complex?

Is the E8 manifold homeomorphic to a CW complex? (I know that it is not triangulable) Edit: The E8 manifold is the unique compact (without boundary), simply connected topological 4-manifold, whose ...
11
votes
1answer
445 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 ...
11
votes
1answer
660 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 ...
11
votes
1answer
289 views

Fractional versions of euclidean space?

This is going to be a somewhat vague question, but I'll be happy if you indulge me. Euclidean space $\mathbb{R}^n$ is equipped with a lot of nice (algebraic, metric, topological,...) structure and ...
10
votes
2answers
396 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 ...
10
votes
2answers
169 views

Partitioning the plane into three sets each intersecting the vertices of every square with side 1?

Q1. Is it possible to partition the plane into three sets such that each of them contains at least one vertex of every square with side 1 ? (I mean all squares of side-length 1, not just those with ...
9
votes
1answer
245 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 ...
9
votes
2answers
237 views

Surgery results in a cylinder

While reading a proof of a theorem about Reshetikhin Turaev topological quantum field theory, I encountered the following problem. Suppose we have several unlinked unknots $K_i$, $i=1, \dots, g$ in ...
9
votes
1answer
662 views

Understanding the Hopf fibration

I'm taking a class in manifolds, and the Hopf fibration recently came up. I'm trying to get a handle on it, so I'm going to try and explain what I think is going on, and hopefully math.stackexchange ...
8
votes
1answer
254 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 ...
8
votes
2answers
119 views

$n$ points on every line

For which integers $n$ is it possible to find a subset $S$ of $\mathbb R^2$ such that every infinite line contains exactly $n$ points of $S$?
8
votes
1answer
584 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 ...
8
votes
1answer
64 views

Parallelizing lines

Let $n \geq 1$ be an integer, and $L_1,\ldots,L_n$ be $n$ lines in $\mathbb{R}^3$ which are pairwise disjoint. Is it possible to move all $n$ lines continuously so that they never cross, and so as to ...
8
votes
1answer
116 views

Playing with the torus and semisimplicial sets (prove that $\phi$ and $\psi$ are not homotopic)

Recall that we can express the torus $|X.| \cong T$ as a square with edges $e$ and $f$, diagonal $g$, faces $T_1$ and $T_2$, and a single vertex $v$, and appropriate identifications. Let $Y.$ be the ...
8
votes
0answers
257 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 ...
7
votes
4answers
620 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 ...
7
votes
3answers
426 views

Putting Geometries on Knot Complements

I have two different, but related, questions about the type of geometry one can get on a knot complement. Quickly some notation: $K$ will be a non-trivial smooth knot - living in $S^3$ - and $M$ will ...
7
votes
1answer
277 views

How to Classify $2$-Plane Bundles over $S^2$?

I'm curious how one can classify the bundles over a given manifold. I recently read this paper on classifying $2$-sphere bundles over compact surfaces. A lot of the concepts went over my head since ...
7
votes
2answers
358 views

Does the Euler characteristic of a manifold depend upon the field of coefficients?

Define the Euler characteristic of a space $X$ to be $$\chi(X)= \sum_i \dim H_i(X, \mathbb Q)$$ This is obviously not necessarily well-defined for an arbitrary space $X$, so let $X$ be a manifold ...
7
votes
2answers
87 views

Are $\{(x,y)\in \mathbb{R}^2 : (x,y)\neq(0,0)\}$ and $\{(x,y)\in \mathbb{R}^2 : (x,y)\notin [0,1]\times\{0\}\}$ homeomorphic?

Let $X_1$ and $X_2$ be the spaces \begin{align*} X_1&=\{(x,y)\in \mathbb{R}^2 : (x,y)\neq(0,0)\}, \\ X_2&=\{(x,y)\in \mathbb{R}^2 : (x,y)\notin [0,1]\times\{0\}\}. \end{align*} Are these ...
7
votes
1answer
327 views

Does there exist homeomorphism without fixed points?

Does there exist a homeomorphism of the unit disk with two holes $$\left\{(x,y):x^2+y^2 \le 1\right\} \setminus \left (\left \{(x,y):\left(x+ \frac 1 2 \right)^2+y^2 < \frac 1 {10} \right \} ...
7
votes
1answer
353 views

Easier proof about suspension of a manifold

For what manifolds $M$ is the suspension $\Sigma M$ also a manifold? By the suspension of a topological space $X$ (not necessarily a manifold), I mean the space $$\Sigma X = (X \times [0,1])/{\sim}$$ ...
7
votes
1answer
130 views

Loop space and stable homotopy theory

The Bott periodicity theorem for unitary group $U(n)$ says that $$ \pi_{i-1}(U) \simeq \pi_{i+1}(U) $$ How can I prove, using this theorem, that $$ \Omega (U) \simeq BU \times \mathbb{Z} ?$$ What is ...
7
votes
1answer
93 views

When $f\colon M\to N\times N$ satisfies that $f^{-1}(\Delta)$ is a ball?

Let $M$, $N$ be smooth manifolds of dimension $m+n$ and $n$, respectively. Suppose that $f\colon M\to N\times N$ is a smooth map and $f$ is transversal to the diagonal $\Delta=\{(x,x)\in N\times ...
7
votes
0answers
62 views

Is “slightly deform” a well defined concept in mathematical proof?

In topological proofs the phrase "slightly deform" is widely used. To me, although I can accept the idea intuitively, the phrase "slightly deform" does not sound like a strict mathematical concept. ...
6
votes
2answers
375 views

Euler characteristic 1: Half a hole?

The Euler characteristic of a two-dimensional disk is $\chi=1$. If one blindly interprets the disk as a closed, orientable surface, then $\chi = 2 - 2g$, and the genus is $g=\frac{1}{2}$. Is there ...
6
votes
1answer
218 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? ...
6
votes
1answer
452 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 ...
6
votes
1answer
246 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 $g$ ...
6
votes
1answer
101 views

Homotopic equivalence for $S^5$ without three $S^1$

Consider a standard embedding of $S^5$ in $\mathbb R^6$: $S^5: \; x_1^2 + x_2^2 + x_3^2 + x_4^2 + x_5^2 + x_6^2 = 1.$ And consider three circles, which are sections of $S^5$ by $x_1 x_2$, $x_3 x_4$, ...