Questions about algebraic methods and invariants to study and classify topological spaces: homotopy groups, (co)-homology groups, and beyond.

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337
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6answers
65k views

Why can you turn clothing right-side-out?

My nephew was folding laundry, and turning the occasional shirt right-side-out. I showed him a "trick" where I turned it right-side-out by pulling the whole thing through a sleeve instead of the ...
85
votes
2answers
3k views

Does a four-variable analog of the Hall-Witt identity exist?

Lately I have been thinking about commutator formulas, sparked by rereading the following paragraph in Isaacs (p.125): An amazing commutator formula is the Hall-Witt identity: ...
77
votes
3answers
8k views

Topology: The Board Game

Edit: I've drawn up some different rules, a map and some cards for playing an actual version of the game. They're available at my personal website with a Creative Commons Attribution 4.0 license. ...
65
votes
2answers
3k views

Direct proof that the wedge product preserves integral cohomology classes?

Let $H^k(M,\mathbb R)$ be the De Rham cohomology of a manifold $M$. There is a canonical map $H^k(M;\mathbb Z) \to H^k(M;\mathbb R)$ from the integral cohomology to the cohomology with coefficients ...
52
votes
1answer
2k views

Simplicial homology of real projective space by Mayer-Vietoris

Consider the $n$-sphere $S^n$ and the real projective space $\mathbb{RP}^n$. There is a universal covering map $p: S^n \to \mathbb{RP}^n$, and it's clear that it's the coequaliser of $\mathrm{id}: S^n ...
46
votes
2answers
2k views

Sheaf cohomology: what is it and where can I learn it?

As I understand it, sheaf cohomology is now an indispensable tool in algebraic geometry, but was originally developed to solve problems in algebraic topology. I have two questions about the matter. ...
41
votes
2answers
1k views

Algebraic Topology Challenge: Homology of an Infinite Wedge of Spheres

So the following comes to me from an old algebraic topology final that got the best of me. I wasn't able to prove it due to a lack of technical confidence, and my topology has only deteriorated since ...
40
votes
6answers
7k views

Hanging a picture on the wall using two nails in such a way that removing any nail makes the picture fall down

A friend of mine told me that it's possible to hang a picture on the wall from a string using two nails in such a way that removing either of the two nails will make both the string and picture fall ...
34
votes
2answers
1k views

How much rigour is necessary?

I am taking a course in Algebraic Topology. We are using Hatcher as a textbook. One of the main problems I am facing with the textbook is its level of rigour. Example: On Pg 10, Hatcher mentions in ...
32
votes
13answers
2k views

How To Present Algebraic Topology To Non-Mathematicians?

I am writing my master thesis in algebraic topology (fundamental groups) and as a system in my school students must write about one page about their theses explaining for non mathematicians the ...
31
votes
5answers
3k views

Soft Question - Intuition of the meaning of homology groups

I'm studying homology groups and I'm looking to try and develop, if possible, a little more intuition about what they actually mean. I've only been studying homology for a short while, so if possible ...
29
votes
10answers
4k views

A really complicated calculus book

I've been studying math as a hobby, just for fun for years, and I had my goal to understand nearly every good undergraduate textbook and I think, I finally reached it. So now I need an another goal. ...
29
votes
2answers
2k views

Which manifolds are parallelizable?

Recall that a manifold $M$ of dimension $n$ is parallelizable if there are $n$ vector fields that form a basis of the tangent space $T_x M$ at every point $x \in M$. This is equivalent to the tangent ...
29
votes
2answers
1k views

How do different definitions of “degree” coincide?

I've recently read about a number of different notions of "degree." Reading over Javier Álvarez' excellent answer for the thousandth time finally prompted me to ask this question: How exactly do ...
29
votes
3answers
409 views

Is $T^1 S^5$ abstractly diffeomorphic to $S^4\times S^5$?

One can ask When is $T^1 S^n$, the unit tangent bundle of $S^n$, abstractly diffeomorphic to $S^{n-1}\times S^n$? For even $n$, the answer is never. This is because $T^1 S^{2n}$ has torsion in ...
29
votes
2answers
677 views

Homology of cube with a twist

Take the quotient space of the cube $I^3$ obtained by identifying each square face with opposite square via the right handed screw motion consisting of a translation by 1 unit perpendicular to the ...
28
votes
3answers
1k views

$\pi_{1}({\mathbb R}^{2} - {\mathbb Q}^{2})$ is uncountable

Question: Show that $\pi_{1}({\mathbb R}^{2} - {\mathbb Q}^{2})$ is uncountable. Motivation: This is one of those problems that I saw in Hatcher and felt I should be able to do, but couldn't quite ...
28
votes
3answers
935 views

Intuition behind Snake Lemma

I've been struggling with this for some time. I can prove the Snake Lemma, but I don't really “understand” it. By that I mean if no one told me Snake Lemma existed, I would not even ...
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 ...
27
votes
3answers
584 views

How did we know to invent homological algebra?

Update: Qiaochu Yuan points out in the comments that the title of the question is misleading, as homological algebra did not begin with long exact sequences as I'd thought. (Original question ...
26
votes
4answers
973 views

Geometry or topology behind the “impossible staircase”

This question on the topology of Escher games reminded me of a question I've had in my head for a little while now. Is there anything interesting geometric or topological that can be said about the ...
25
votes
1answer
1k views

Is this space contractible?

Let $X$ be the following topological space (with the subspace topology): Connect the rational points of $([0,1]\cap \mathbb{Q})\times \{0\}$ with the point $(0,1)$ and connect the points of ...
25
votes
1answer
889 views

Perturbation trick in the proof of Seifert-van-Kampen

The theorem of Seifert-Van-Kampen states that the fundamental group $\pi_1$ commutes with certain colimits. There is a beautiful and conceptual proof in Peter May's "A Concise Course in Algebraic ...
24
votes
4answers
3k views

Learning Roadmap for Algebraic Topology

I am now considering about studying algebraic topology. There are a lot of books about it, and I want to choose the most comprehensive book among them. I have a solid background in Abstract Algebra, ...
24
votes
3answers
1k views

Roadmap to study Atiyah-Singer index theorem

I am a physics undergrad and want to pursue a PhD in Math (geometry or topology). I study it almost completely by myself, as the program in my country offers very less flexibility to take non ...
23
votes
2answers
901 views

What is the topology of a world with portals?

Portal is a video game, where you can create 2 disks $D\in\mathbb{R}^3$, which then are identified. The world is glued together at these points. This kind of reminds me of some procedures to ...
23
votes
0answers
571 views

Is there a homology theory that counts connected components of a space?

It is well-known that the generators of the zeroth singular homology group $H_0(X)$ of a space $X$ correspond to the path components of $X$. I have recently learned that for Čech homology the ...
22
votes
3answers
1k views

is the group of rational numbers the fundamental group of some space?

Which path connected space has fundamental group isomorphic to the group of rationals? More generally, is every group the fundamental group of a space?
22
votes
3answers
2k views

Research in algebraic topology

I have started studying algebraic topology with the help of Armstrong(Basic), Massey, and Hatcher. If I plan to do research in algebraic topology in future: What else should I study after ...
22
votes
2answers
267 views

Paracompactness of CW complexes (rather long)

I finished reading Lee's 'introduction to topological manifolds' (2nd edition) and I'm currently tying up some loose ends. One thing I can't understand is the proof of paracompactness of CW complexes. ...
22
votes
1answer
325 views

Does every continuous map from $\mathbb{H}P^{2n+1}$ to itself have a fixed point?

This question is motivated by Fixed point property of Cayley plane and idle curiosity. In the link, it is shown that every continuous map from the Cayley plane to itself has a fixed point and the ...
21
votes
3answers
1k views

Consequences of Degree Theory

I'm preparing a presentation on an overview of algebraic and differential topology, and my introduction includes some motivational material on Degree Theory. I have two fundamental and invaluable ...
21
votes
3answers
1k views

Why algebraic topology is also called combinatorial topology?

I remember reading somewhere(at least more than once) that algebraic topology is also known by the name "Combinatorial Topology" which essentially tags the subject fundamentally with some counting ...
21
votes
2answers
310 views

How can I lift a path to $\mathrm{Spin}(n)$?

Suppose I am given an explicit differentiable path $\gamma\colon[a,b]\to SO(n)$, with $\gamma(a)=\gamma(b)=I$. Then $\gamma$ either does or does not lift to a closed loop in $\mathrm{Spin}(n)$. How ...
21
votes
1answer
517 views

An interesting topological space with $4$ elements

There is an interesting topological space $X$ with just four elements $\eta,\eta',x,x'$ whose nontrivial open subsets are $\{\eta\},\{\eta'\},\{\eta,\eta'\}, \{\eta,x,\eta'\}, \{\eta,x',\eta'\}$. This ...
21
votes
1answer
374 views

How much of an $n$-dimensional manifold can we embed into $\mathbb{R}^n$?

I observed some naive examples. Spheres, for example, when we cut out one point, can be embedded into $\mathbb{R}^n$. And if we cut out a measure zero set of a projective space, it can be embedded ...
21
votes
1answer
460 views

Shrinking Group Actions

Suppose $H\subset G$ is a subgroup of a topological group $G$, and $Y\subset X$ is a subspace of a topological space $X$. Suppose we are given a continuous group action $\rho : G\times X\rightarrow X$ ...
20
votes
1answer
1k views

Cohomology of projective plane

How I can compute cohomology de Rham of the projective plane $P^{2}(\mathbb{R})$ using Mayer vietoris or any other methods?
20
votes
1answer
797 views

The Fundamental group of every subset of $\mathbb{R^2}$ is torsion free?

It seems that the fundamental group of any subset of $\mathbb{R^2}$ will not have an element of finite order. Though the 3-dimensional version is an open problem I couldn't immediately see why it is ...
20
votes
2answers
216 views

Confusion about covering projection

This is quite a detailed question: I'm struggling to understand a few parts of a proof of the following Lemma. I've placed stars ($\bigstar$) where I'd like to draw your attention. Lemma: Let ...
19
votes
4answers
897 views

Homotopy groups of $S^2$

I'd like to understand higher homotopy groups better and I guess there's no simpler way than understanding them for as simple spaces as possible; therefore $S^2$. My question essentially has two ...
19
votes
2answers
262 views

Curious remark of D. Ravenel

In his beautiful (but difficult) book "Complex cobordism and stable homotopy groups of spheres", concerned mostly with methods of computing homotopy groups of spheres, D. Ravenel describes a general ...
19
votes
2answers
372 views

How do you define the “boundary” of a topological space?

As described here (and as I always thought was the most general definition of boundary), a possible definition of the boundary of a subset $S$ of a topological space $X$ is $\partial S = \overline S ...
19
votes
1answer
413 views

A homotopy sphere

My question is part of an exercise in Hatcher's 'Algebraic Topology'. Consider a CW complex $X$, constructed from a circle and two 2-disks $e_2$ and $e_3$, attached to that circle by maps of degree 2 ...
18
votes
5answers
850 views

Covering spaces - why are they useful?

As someone who trained as a physicist, I have known for ages that $\operatorname{SU}(2)$ is a double cover of $\operatorname{SO}(3)$, so, during an idle day at the office I decided to look up what ...
18
votes
2answers
544 views

Why isn't $\mathbb{CP}^2$ a covering space for any other manifold?

This is one of those perhaps rare occasions when someone takes the advice of the FAQ and asks a question to which they already know the answer. This puzzle took me a while, but I found it both simple ...
18
votes
3answers
715 views

Is there a gap in the standard treatment of simplicial homology?

On MO, Daniel Moskovich has this to say about the Hauptvermutung: The Hauptvermutung is so obvious that it gets taken for granted everywhere, and most of us learn algebraic topology without ever ...
18
votes
2answers
288 views

Failure of excision for $\pi_2$

Would anyone know an example of failure of excision for 2nd homotopy groups? Specifically, I am looking for $A,B$ open in $X$ such that $X=A\cup B$ and $A\cap B$ is connected and $\pi_2(X,A)\ne ...
18
votes
1answer
1k views

What is Cech homology?

I got into a discussion today about how, just as with all the other ways of computing singular homology, there should be an internal sort of integration pairing for Cech cohomology with "Cech ...
18
votes
1answer
491 views

The simplest nontrivial (unstable) integral cohomology operation

By an integral cohomology operation I mean a natural transformation $H^i(X, \mathbb{Z}) \times H^j(X, \mathbb{Z}) \times ... \to H^k(X, \mathbb{Z})$, where we restrict $X$ to some nice category of ...