# Tagged Questions

69 views

### Main goal of algebraic topology and progress towards the main goal… [closed]

In algebraic topology we use algebraic structure such as group to classify some topological spaces. To do that mathematician invented homotopy group,homology,co-homology etc.One can apply these tools ...
35 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 ...
432 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 ...
101 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 ...
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 ...
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: ...
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
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 ...
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: ...
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 ...
956 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 ...
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 ...
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 ...
487 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: ...
782 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 ...
861 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 ...
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 ...
349 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 PoincarĂ© duality is ...
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 ...