3
votes
1answer
36 views

Intuition behind prism operators to prove homotopy invariance of homology

I'm trying to understand the proof of homotopy invariance of induced maps on homology. However, I do not really understand the intuition behind this proof and especially what the prism operators (as ...
3
votes
2answers
436 views

Fundamental group of projective plane is $C_{2}$???

I just recently know that there are topology with finite nontrivial fundamental group (homotopy curve). I just can't wrap my mind around it at all. If you have a curve, and somehow cannot shrunk it ...
1
vote
2answers
105 views

Intuition behind definition of homotopic equivalence and distinction with homeomorphism

I am a physics student and have come across the definition of homotopic equivalence of two spaces as existence of two functions $f:X \to Y,g: Y \to X$ such that $g \circ f$ and $f \circ g$ are ...
1
vote
1answer
101 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 ...
1
vote
0answers
100 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
170 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
7
votes
2answers
1k 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 ...
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: ...
5
votes
1answer
198 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 ...
29
votes
3answers
963 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
380 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 ...
4
votes
2answers
1k 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 ...
15
votes
2answers
489 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: ...
13
votes
1answer
783 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 ...
18
votes
5answers
866 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
194 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 ...
6
votes
1answer
350 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 ...
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 ...
5
votes
1answer
841 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
368 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
267 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 ...
9
votes
5answers
506 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 $n$, ...