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

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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 ...
22
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, ...
25
votes
5answers
2k 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 ...
21
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?
6
votes
4answers
603 views

Two CW complexes with isomorphic homotopy groups and homology, yet not homotopy equivalent

A standard example of two CW complexes which have isomorphic homotopy groups but are not homotopy equivalent is $ RP^2 \times S^3$ and $RP^3 \times S^2$. The easiest way to see that they are not ...
7
votes
7answers
1k views

How can I prove formally that the projective plane is a Hausdorff space?

I want to prove the Hausdorff property of the projective space with this definition: the sphere $S^n$ with the antipodal points identified. It's seems easy, but I can't prove formally with this ...
318
votes
6answers
64k 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 ...
23
votes
3answers
943 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 ...
27
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 ...
32
votes
3answers
3k 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 $A = \{(U_{\alpha}, \phi_{\alpha})\}$ ...
17
votes
2answers
511 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 ...
12
votes
2answers
231 views

Torsion on $\pi_1(X)$, $X$ connected and open in $\mathbb{R}^n$

Can the fundamental group of an open connected subset $X$ of $\mathbb{R}^n$ have a torsion element?
8
votes
2answers
750 views

Topological group: Multiplying two loops is homotopic to linking these paths?

Let G be a topological group and let $s_1$ and $s_2$ be loops in G (both loops are based at the identity e of G). Is it true that the loop $s_1s_2$ (where the multiplication is the one of the group ...
24
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 ...
11
votes
2answers
294 views

A comparison between the fundamental groupoid and the fundamental group

Are there two path connected topological spaces $X,Y$ such that the fundamental groupoid of $X$ is not isomorphic to the fundamental groupoid of $Y$ but the fundamental group of $X$ is isomorphic to ...
10
votes
5answers
327 views

$\varphi$ in $\operatorname{Hom}{(S^1, S^1)}$ are of the form $z^n$

I'd like to see a proof why $\varphi \in \operatorname{Hom}{(S^1, S^1)}$ looks like $z^n$ for an integer $n$. At first I thought I could argue that if I have a homomorphism that maps $e^{ix}$ to some ...
7
votes
1answer
827 views

Contractible vs. Deformation retract to a point.

I have a quick question about the difference between the two concepts in the title. The question is basically ex.6 (b) in Hatcher's book titled "Algebraic Topology". Let $X$ be the subspace of $R^2$ ...
7
votes
1answer
2k views

Homology of surface of genus $g$

This is a homework question given to me by someone of the community here and it's a generalisation of this. I was wondering if you could have a look and tell me if it's right. Thanks for your help! ...
6
votes
1answer
218 views

Is a CW complex, homeomorphic to a regular CW complex?

Is a CW complex, homeomorphic to a regular CW complex ? Regular means that the attaching maps are homeomorphisms (1-1). In particular, an open (resp. closed) $n$-cell, is homeomorphic to an open ...
6
votes
0answers
1k views

The simply connected coverings of two homotopy equivalent spaces are homotopy equivalent

This is exercise 1.3.8 in Hatcher: Let $\tilde{X}$ and $\tilde{Y}$ be simply-connected covering spaces of path connected, locally path-connected spaces $X$ and $Y$. Show that if $X\simeq Y$ then ...
4
votes
2answers
263 views

Hatcher Ch.0 (P18) #5 Inclusion Map is Nullhomotopic

Question: Show that if a space $X$ deformation retracts to a point $x ∈ X$, then for each neighborhood $U$ of $x$ in $X$ $\exists$ a neighborhood $V ⊂ U$ of $x$ such that the inclusion map $V ...
3
votes
1answer
361 views

Spaces with equal homotopy groups but different homology groups?

Since it's fairly easy to come up with a two spaces that have different homotopy groups but the same homology groups ($S^2\times S^4$ and $\mathbb{C}\textrm{P}^3$). Are there any nice examples of ...
3
votes
2answers
1k views

Surface of genus $g$ does not retract to circle (Hatcher exercise)

I'm trying exercise 9 on page 53 in Hatcher but I need some help with it. The exercise is: In the surface $M_g$ of genus $g$, let $C$ be a circle that separates $M_g$ into two compact subsurfaces ...
1
vote
2answers
909 views

Showing that an inclusion is null homotopic

I'm trying to do exercise 5 on page 18 in Hatcher: Show that if a space $X$ deformation retracts to a point $x \in X$, then for each neighborhood $U$ of $x$ in $X$ there exists a neighborhood $V ...
27
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 ...
18
votes
2answers
266 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 ...
13
votes
1answer
347 views

Does $gHg^{-1}\subseteq H$ imply $gHg^{-1}= H$?

Let $G$ be a group, $H<G$ a subgroup and $g$ an element of $G$. Let $\lambda_g$ denote the inner automorphism which maps $x$ to $gxg^{-1}$. I wonder if $H$ can be mapped to a proper subgroup of ...
6
votes
4answers
6k views

Best Algebraic Topology book/Alternative to Allen Hatcher free book?

Allen Hatcher seems impossible and this is set as the course text? So was wondering is there a better book than this? It's pretty cheap book compared to other books on amazon and is free online. ...
12
votes
2answers
491 views

For every $k \in {\mathbb Z}$ construct a continuous map $f: S^n \to S^n$ with $\deg(f) = k$.

Suppose $S^n$ is an $n$-dimensional sphere. Definition of the degree of a map: Let $f:S^n \to S^n$ be a continuous map. Then $f$ induces a homomorphism $f_{*}:H_n(S^n) \to H_n(S^n)$ . Considering the ...
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?
13
votes
3answers
2k views

What is a natural isomorphism?

I came across the adjective "natural" many times in my reading and I think this has something to do with category theory. Could someone please illustrate the idea behind this adjective to me? Many ...
6
votes
2answers
617 views

Why is every discrete subgroup of a Hausdorff group closed?

I have just began to learn about topological group recently and is still not familiar with combining topology and group theory together. I have read an useful property of discrete group on the ...
5
votes
3answers
576 views

How to compute homotopy classes of maps on the 2-torus?

Let $\mathbb T^2$ be the 2-Torus and let $X$ be a topological space. Is there any way of computing $[\mathbb T^2,X]$, the set of homotopy class of continuous maps $\mathbb T^2\to X$ if I know, for ...
6
votes
2answers
660 views

Unit sphere in $\mathbb{R}^\infty$ is contractible?

Let $\mathcal{T}_{\infty}= \left\{ U \subset \mathbb{R}^{\infty}: \ U \cap \mathbb{R}^n \in \mathcal{T}_n, \text{ for } n=1,2,... \right\} $. Of course $\mathcal{T}_{\infty}$ is topology in ...
4
votes
2answers
786 views

Why is this entangled circle not a retract of the solid torus?

I'm doing exercise 16 on page 39 in Hatcher: Show that there are no retractions $r: X \rightarrow A$ in the following cases: (a) $X = \mathbb{R}^3$ with $A$ any subspace homeomorphic to $S^1$ (b) ...
5
votes
1answer
475 views

weak homotopy equivalence (Whitehead theorem) and the *pseudocircle*

On wikipedia, I recently read about a highly pathological finite topological space, namely the pseudocircle $$X=\{a,b,c,d\},\;\;\; \mathcal{T}=\{\emptyset,\{a\},\{b\},\{ab\},\{a,b,c\},\{a,b,d\},X\}.$$ ...
3
votes
4answers
672 views

Why this map is a covering map?

I'm trying to find the universal covering space of the Klein bottle. I know that $\mathbb R^2$ covers the Klein bottle , but I don't know how to prove, I found this proof on internet: Someone knows ...
8
votes
1answer
215 views

The complement of Jordan arc

If $A$ is the image of a Jordan arc in $S^2$, that is, $A$ is the image of an injective continuous map from $[0,1]$ to $S^2$, is $S^2-A$ necessarily a simply-connected set?
4
votes
2answers
229 views

About covering maps and sections!

If $q: E\rightarrow X$ is a covering map that has a section $(i.e. f: X\rightarrow E, q\circ f=Id_X)$ does that imply that $E$ is a 1-fold cover?
3
votes
1answer
200 views

Automorphism Group of a graph

If $X$ is a locally finite graph, (i.e. each vertex has finite index), is it true that the automorphism group Aut($X$) of the graph X is locally compact? Here, Aut($X$) has compact open topology; and ...
2
votes
2answers
117 views

Question on relative homology

I have this question and I'd like an idea to solve it: If $Z_p(X,Y)=\lbrace \sigma\in C_p(X), \partial\sigma\in C_{p-1}(Y)\rbrace$, $1)$prove that $H_p(X,Y)$ is isomorphic to ...
43
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. ...
26
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 ...
13
votes
4answers
2k views

What algebraic topology book to read after Hatcher's?

I've currently finished chapter 2 of his book and done all the exercises of in chapter 0, 1 and 2. Was wondering when I finished reading this book what book do I read next in algebraic topology?
20
votes
1answer
770 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
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 ...
15
votes
3answers
1k views

Fundamental group of GL(n,C) is isomorphic to Z. How to learn to prove facts like this?

I know, fundamental group of $GL(n,\mathbb{C})$ is isomorphic to $\mathbb{Z}$. It's written in Wikipedia. Actually, I've succeed in proving this, but my proof is two pages long and very technical. I ...
14
votes
4answers
2k views

Good exercises to do/examples to illustrate Seifert - Van Kampen Theorem

I have just learned about the Seifert-Van Kampen theorem and I find it hard to get my head around. The version of this theorem that I know is the following (given in Hatcher): If $X$ is the ...
9
votes
4answers
868 views

Poincare Duality Reference

In Hatcher's "Algebraic Topology" in the Poincare Duality section he introduces the subject by doing orientable surfaces. He shows that there is a dual cell structure to each cell structure and it's ...
7
votes
2answers
1k views

Proof of another Hatcher exercise: homotopy equivalence induces bijection

I'm doing stuck with the first half of exercise 12 on page 19 in Hatcher: Exercise: Show that a homotopy equivalence $f : X \rightarrow Y$ induces a bijection between the set of path-components of ...