Questions about algebraic methods and invariants to study and classify topological spaces: homotopy groups, (co)-homology groups, and beyond.
273
votes
6answers
60k 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 ...
64
votes
2answers
6k views
Topology: The Board Game, plus Clue: Topology Edition
5/25/13 Edit: I realized today that this game can be adapted as a fun reworking of the board game Clue. Instead of rolling a die, each player draws a topology card, then moves according to the ...
56
votes
2answers
2k 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 ...
34
votes
2answers
811 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 ...
29
votes
2answers
804 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 ...
28
votes
3answers
2k views
Why is the Möbius strip not orientable?
I am trying to understand the notion of an orientable manifold.
Let M be a smooth n-manifold. We say that M is orientable if and only if there exists an atlas $A = \{(U_{\alpha}, \phi_{\alpha})\}$ ...
27
votes
0answers
1k 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: ...
27
votes
0answers
1k 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 ...
25
votes
1answer
1k 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.
...
25
votes
1answer
439 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 ...
24
votes
3answers
312 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 ...
23
votes
3answers
658 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 ...
22
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 ...
21
votes
9answers
3k 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. ...
21
votes
3answers
1k 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 ...
21
votes
2answers
1k 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 ...
20
votes
2answers
609 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 ...
20
votes
1answer
885 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 ...
19
votes
3answers
508 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 ...
19
votes
1answer
605 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 ...
19
votes
1answer
387 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 ...
18
votes
3answers
748 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 ...
18
votes
2answers
174 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 ...
17
votes
3answers
691 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?
17
votes
5answers
1k 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 ...
17
votes
2answers
156 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. ...
17
votes
1answer
621 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 ...
16
votes
3answers
814 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 ...
16
votes
3answers
835 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 ...
16
votes
4answers
2k 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, ...
16
votes
3answers
604 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 ...
16
votes
2answers
323 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 ...
16
votes
3answers
792 views
Motivating Cohomology
Question: Are there intuitive ways to introduce cohomology? Pretend you're talking to a high school student; how could we use pictures and easy (even trivial!) examples to illustrate cohomology?
Why ...
16
votes
1answer
216 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 ...
16
votes
1answer
879 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 ...
16
votes
1answer
617 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 ...
16
votes
1answer
225 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 ...
16
votes
0answers
274 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 ...
15
votes
2answers
435 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 ...
15
votes
4answers
286 views
What are the ramifications of the fact that the first homotopy group can be non-commutative, whilst the higher homotopy groups can't be?
Does this mean that the first homotopy group in some sense contains more information than the higher homotopy groups? Is there another generalization of the fundamental group that can give rise to ...
15
votes
2answers
3k views
“A proof that algebraic topology can never have a non self-contradictory set of abelian groups” - Dr. Sheldon Cooper
In the current episode "The Big Bang Theory", Dr. Sheldon Cooper has a booklet titled "A proof that algebraic topology can never have a non self-contradictory set of abelian groups". I'm still an ...
15
votes
4answers
701 views
Advanced algebraic topology topics overview
Recently I became very much intrigued by algebraic topology and am spending quite some time learning it. My reasons are three-fold:
it's a beautiful theory;
it gives geometric justification to (or ...
15
votes
2answers
816 views
How useless can the Mayer-Vietoris sequence be in general?
In an algebraic topology course I'm taking we are often asked to compute the homology groups of a space $X = A \cup B$ using the Mayer-Vietoris sequence, and it happens in all of the examples I've ...
15
votes
1answer
818 views
Presentation of the fundamental group of a manifold minus some points
I recently noticed a few things in some recent questions on MO:
1) the fundamental group of $S^2$ minus, say, 4 points, is $\langle a,b,c,d\ |\ abcd=1\rangle$.
2) The fundamental group of a torus ...
15
votes
1answer
262 views
Mapping class group vs outer automorphism group of the fundamental group for nonorientable surfaces
The Dehn--Nielsen--Baer theorem states that for a closed, connected and orientable surface M the extended mapping class group of M is isomorphic to the outer automorphism group of the fundamental ...
15
votes
2answers
1k views
Visualizing the fundamental group of SO(3)
Recently I became interested in trying to visualize the fact that $\pi_1(\text{SO}(3)) = \mathbb{Z}/2\mathbb{Z}$. For whatever reason, the plate trick doesn't do it for me, so I've been looking for ...
15
votes
1answer
236 views
Twisted Cech cohomology
Let $X$ be a CW-complex with contractible universal cover $\tilde{X}$ and fundamental group $\pi = \pi_1X$. Twisted (co)homology is found by lifting the cell structure on $X$ to a $\pi$-invariant ...
15
votes
1answer
185 views
Geometric interpretation of the map $SO(4) \to SO(3)$
Let me first explain the background of my question.
As is well known, the group $SO(n+1)$ acts transitively on the sphere $S^n$, and the stabilizer is the group $SO(n)$, so that we get a fibration ...
15
votes
1answer
142 views
Closure by Projective Limits of the category of Coverings of a Topological Space
Let $X$ be a connected topological space, and $C_{finite}$ the
category of its finite coverings. Then I claim that the category $C$
of coverings of $X$ can be obtained by $C_{finite}$ taking
...
14
votes
5answers
576 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 ...
