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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99
votes
3answers
6k 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 ...
70
votes
1answer
2k 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. ...
58
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 $$\...
34
votes
3answers
528 views

Does every bijection of $\mathbb{Z}^2$ extend to a homeomorphism of $\mathbb{R}^2$?

Given a bijection $f\colon \mathbb{Z}^2 \to \mathbb{Z}^2$, does there always exist a homeomorphism $h\colon\mathbb{R}^2\to\mathbb{R}^2$ that agrees with $f$ on $\mathbb{Z}^2$? I don't see any ...
34
votes
6answers
716 views

Is the complement of countably many disjoint closed disks path connected?

Let $\{D_n\}_{n=1}^\infty$ be a family of pairwise disjoint closed disks in $\mathbb{R}^2$. Is the complement $$ \mathbb{R}^2 -\bigcup_{n=1}^\infty D_n $$ always path connected? Here “...
27
votes
4answers
3k 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 ...
20
votes
2answers
957 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. ...
19
votes
1answer
474 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 ...
18
votes
4answers
874 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 S^{...
18
votes
1answer
364 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 ...
17
votes
2answers
295 views

What closed 3-manifolds have fundamental group $\Bbb Z$?

For certain small groups, it is easy (and desirable) to classify closed (and orientable if necessary) 3-manifolds with that group as their fundamental group. (Essentially due to Waldhausen is that for ...
17
votes
1answer
495 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 R^{n-...
16
votes
2answers
371 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 ...
16
votes
2answers
894 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 ...
15
votes
1answer
273 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 ...
14
votes
3answers
457 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 ...
14
votes
1answer
384 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 components....
13
votes
4answers
2k 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 ...
13
votes
1answer
324 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
1answer
681 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
1k 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
422 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 ...
11
votes
1answer
287 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
177 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
401 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
214 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
503 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
388 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
4answers
925 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 ...
10
votes
2answers
546 views

Why are invariants of knots and manifolds important or useful?

It is easy to define invariants that completely classify knots; however, this is computationally infeasible, so is it computationally efficient invariants that are important? Why? Do mathematicians ...
10
votes
2answers
4k views

Understanding the Equation of a Möbius Strip

I am in HL Math and trying to finish my IA. My topic is the Möbius band. The only problem is, I do not understand the formula that defines it and everywhere I have looked has just given me a math-...
10
votes
2answers
265 views

Which groups act freely on $S^n$?

When $n$ is even, it is easy to classify groups which act freely on $S^n$ using degree theory: if $G$ acts on $S^n$, then associating to each element $g \in G$ the degree of the map obtained from ...
10
votes
2answers
410 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 $0\!<\!\varepsilon\!<\!1$...
10
votes
1answer
280 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 ...
10
votes
2answers
190 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
2answers
606 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 (...
9
votes
1answer
95 views

Two disjoint real projective planes in real projective space?

Let $\mathbb{R}\mathbb{P}^3$ be the real projective three-space. It is clear that any two hyperplanes in $\mathbb{R}\mathbb{P}^3$ intersect. But I wonder whether one could embed two copies of the real ...
9
votes
1answer
597 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 ...
9
votes
1answer
164 views

General relationship between braid groups and mapping class groups

I just finished correcting my answer on visualizing braid groups as fundamental groups of configuration spaces, and in the process became interested in the other pictorial definition of the braid ...
9
votes
1answer
120 views

What are the 8 non-compact Euclidean 3-manifolds?

I have found several sources that mention that there are eight non-compact Euclidean 3-manifolds, with four orientable and four non-orientable. There are two standard references for this, but ...
9
votes
1answer
598 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 $$\Sigma X = (X \times [0,1])/{\sim}$$ ...
9
votes
1answer
78 views

In how many dimensions is the full-twisted “Mobius” band isotopic to the cylinder?

There is a question on this site about the distinctions between the full-twisted Mobius band and the cylinder, but I would like to ask something different, so I start a new question. Let us call $C$ ...
9
votes
2answers
246 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
804 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 ...
9
votes
0answers
166 views

Prerequisites for studying Perelman's proof of the Geometrization Conjecture

I want to set a course toward understanding Perelman's proof of the Geometrization Conjecture. I realize this will be a lengthy undertaking, but hopefully only on the order of one to two years. I am ...
9
votes
0answers
321 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 ...
8
votes
3answers
455 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 ...
8
votes
2answers
544 views

Length of the main diagonal of an n-dimensional cube

Find the length of a main diagonal of an n-dimensional cube, for example the one from $(0,0,...,0)$ to $(R,R,...,R)$ I tried to use induction to prove that its $\sqrt{n}R$ but I'm stuck on writing ...
8
votes
1answer
467 views

Equivalence of knots: ambient isotopy vs. homeomorphism

I am looking into knot theory and have found two different definitions stating that two knots $K_1$ and $K_2$ are equivalent, namely the concept of an ambient isotopy: These two knots are ambient ...
8
votes
1answer
118 views

What is this manifold?

As picture below ,it is a Mobius band with a cylinder crossing it .Let it be $\Omega$ . Obviously , $\partial \Omega$ is a circle. Now , what is $\Omega/\partial \Omega$ ( I mean glue the boundary to ...