Tagged Questions

1
vote
1answer
37 views

Weak homotopy equivalence

I know a continuous map $f:X\to Y$ between topological spaces is a weak homotopic equivalence if it induces isomorphisms on the corresponding homotopy groups, but what kind of information do I get ...
0
votes
0answers
17 views

Understanding the topology of a variety concretely

My ultimate goal is to understand how to compute the cohomology groups of complex algebraic varieties, without having to know what a scheme is. Therefore I want to be able to handle simple examples, ...
1
vote
0answers
87 views

Which space this space drawn in this picture is homeomorphic?

Based in this question Why this space is homeomorphic to the plane? I would like to know which space this space is homeomorphic, any help or an intuitive idea are welcome. [Context of Image: ...
3
votes
1answer
132 views

Why this space is homeomorphic to the plane?

I'm trying to see why this picture below is homeomorphic to the $\mathbb R^2$. It's really hard, please I need an intuitive idea of this. This seems very weird for me, I need help. Thanks a lot
6
votes
2answers
870 views

This quotient space is homeomorphic to the Möbius strip?

Let $G:\mathbb R \times [-1,1]\to \mathbb R \times [-1,1]$ be a map defined by $G(x,y)=(x+1,-y)$ This space $Q=\mathbb R\times [-1,1]/\sim$, where $(x_1,y_1)\sim (x_2,y_2)$ if and only if there is ...
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: ...
4
votes
1answer
98 views

Intuition for cofibration

The notion of a fibration has a nice geometric intuition of one topological space (a fiber) being parametrized by another topological space (the base) -- this is taken from the Wikipedia entry on ...
19
votes
3answers
505 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 ...
3
votes
2answers
292 views

$SO(3)$ Lie group

I'm a little stuck at the moment where to go next with this. I know that there is a fact that there is a curve in $SO(3)$, beginning and ending at the identity which cannot be deformed to the constant ...
3
votes
2answers
675 views

Find all connected 2-sheeted covering spaces of $S^1 \lor S^1$

This is exercise 1.3.10 in Hatcher's book "Algebraic Topology". Find all the connected 2-sheeted and 3-sheeted covering spaces of $X=S^1 \lor S^1$, up to isomorphism of covering spaces without ...
11
votes
2answers
242 views

Can we think of a chain homotopy as a homotopy?

I'm taking a course in algebraic topology, which includes an introduction to (simplicial) homology, and I'm looking for a bit of intuition regarding chain homotopies. The definitions I'm using are: ...
10
votes
1answer
442 views

Intuition behind homology with general coefficients

We just went over homology with general coefficients in topology and did some of the usual examples ($\mathbb{Z}_2$ for projective space and manifolds being the big examples) which led me to wonder ...
14
votes
5answers
575 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 ...
4
votes
1answer
142 views

Visualizing homologous elements

For the fundamental group it's easy to visualize when two loops are homotopic. I was wondering if there are any ways to look at the equivalent problem for homology? I guess this might be tricky for ...
5
votes
1answer
187 views

Intuition for Poincaré duality and Cap product

Can you provide me with any intuition behind the Cap product of a cohomology class and a homology class? What is its geometric meaning? Can you also give me an intuition why the Poincaré duality is ...
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 ...
5
votes
1answer
550 views

The simplest $\Delta$-complex structure on $S^2$

I think my reasoning is correct, but I want to run through it here because having the right intuition will make similar problems easier in future. A 2-simplex is homeomorphic to a closed disc, and a ...
8
votes
1answer
262 views

The Mayer-Vietoris sequence

If $X$ is a space with a pair of subspaces $A, B \subset X$ such that $X$ is the union of the interiors of $A$ and $B$, then there is a long exact sequence of homology groups $\displaystyle \ldots\to ...
9
votes
2answers
238 views

geometric argument for van-kampen?

I've seen Van-Kampen's theorem presented algebraically many times; and although it provides a useful method of calculation; I don't have a very clear picture for "why" it should be true. Does anyone ...
8
votes
5answers
444 views

Visualising $\mathbb CP^2$: a problem of attaching cells with a dimension gap >1

For the uninitiated Morse theory, as many other early alebraic-topology widgets, leads to a picture of smooth manifolds as being built up from 'cells', copies of $\mathbb{D}^n$ for varying $n$, ...