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

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4
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1answer
146 views

Moore Spaces: explicit CW-complex for $M(\mathbb{Z}_m, n)$

Given an abelian group $G$ and an integer $n \ge 1$ we can construct a $CW$ complex such that $H_n(X) \cong G$ and $\tilde{H}_i(X)=0$ for all $i \neq n$. We call this $CW$ complex a Moore space and ...
1
vote
1answer
36 views

Where are the vertices of the universal cover of the genus 2 torus octagon?

The universal cover of the genus 2 torus is hyperbolic plane and the fundamental domain is a octagon. Here is a picture, which I took from here. Is there a closed form for the points of set of the ...
2
votes
0answers
65 views

Difference between algebraic topology and geometric topology [closed]

What are the main differences between these two areas? Does geometric topology in general, use more analytic techniques? Which one would most consider harder? Is one more general than the other?
1
vote
2answers
146 views

What exactly are the elements of a local homology group?

A local homology group of some space $X$ at $x \in X$ is defined by the relative homology group $H_n(X, X - x)$. So by definition, it contains only cycles that are not entirely contained in $X - x$. ...
8
votes
0answers
124 views

Open questions in Topological K-Theory

I am interested in knowing about current research in the Topological K-Theory, especially its interactions with String Theory. About one and a half decade back, there were some papers by Physicists (e....
0
votes
1answer
26 views

How to show graphs are homeomorphic (or not)

Hi, I am trying to figure out which of these are homeomorphic (topologically equivalent) I know that number of cut points, and numver of vertices (where degree is ont 2) are topological invariants ...
2
votes
1answer
60 views

Is there any general method to calculate the universal cover of a given topological space $X$?

I am currently taking a course in algebraic topology and I have to calculate the universal cover of a lot of spaces (I'll call them $X$ from here on). So far I know some tricks to do it: consider $X'$ ...
2
votes
1answer
33 views

Homology of pairs

This shows up as problem 2.2.26 in Hatcher's Algebraic Topology. Given a pair $(X,A)$ let $X\cup CA$ be $X$ with a cone $CA$ attached at $A$. Suppose that $A$ contractible in $X$. I want to show $...
4
votes
1answer
93 views

How can I prove that the hawaiian earring has no universal cover?

I know that the Hawaiian earring is not semi-locally simply connected so the existence is not guaranteed. Also, the point in which it must fail is the origin, where it isn't even locally simply ...
2
votes
0answers
116 views

Orientable cover of a non-orientable manifold factored through the orientation double cover.

While proving that orientable cover $M$ of a manifold non-orientable manifold $N$ factored through the orientation double cover, I got stuck in this following problem... If $p:N→M$ is a covering, N ...
0
votes
1answer
28 views

Explicit homotopy between $f:S^1\to S^1$ the antipodal map and the identity map

I know that $f$ is rotating $180$ degrees and the identity is rotating by $0$ degrees. How do I write down an explicit homotopy between these two maps? I know this is a stupid question but I can't ...
3
votes
1answer
86 views

Covering of hawaiian earring

I'm taking a course on Algebraic Topology and I'm struggling to find the solution to this problem: Let $Y$ be the Hawaiian earring in $\mathbb{R}^2$ and $Y'$ the union on infinite $Y$s moved $3z$ ...
0
votes
1answer
15 views

Example of non locally connected space with a covering in each connected component which is not a covering of the whole space

Can someone show me an example of an space $X$ non locally connected and another space $X'$ such that $\varpi: X' \to X$ is not a covering but $\varpi: \varpi^{-1}(X_i) \to X_i$ is, for each connected ...
1
vote
1answer
44 views

Is $H_n(X)$ just a different way of writing$H_n(S_*(X))$?

While studying homology in algebraic topology, I sometimes see the notation $H_n(S_*(X))$, and sometime the notation $H_n(X)$. I think these are supposed to be the same, but I'm not sure. The first ...
1
vote
1answer
35 views

Obtaining the Fundamental Polygon of $\mathbb{R}P^2$

On this page, Wikipedia shows, under the "Examples" heading, the fundamental polygons of the Sphere and the Real Projective Plane. Can we obtain the latter diagram from the former? I thought that this ...
1
vote
2answers
151 views

Intuitive or visual understanding of the real projective plane

If we take the definition of a real projective space $\mathbb{R}\mathrm{P}^n$ as the space $S^n$ modulo the antipodal map ($x\sim -x$), it is possible to see that $\mathbb{R}\mathrm{P}^1$ is ...
1
vote
1answer
56 views

Description of normal subgroup when using Seifert van Kampen

I am doing an exercise where I am supposed to compute the fundamental group of $\mathbb{S}^1\times[0,1]$ using Van Kampen's theorem with the open cover $A=\mathbb{S}^1\times[0,3/4)$ and $B=\mathbb{S}^...
1
vote
1answer
55 views

Suppose $C$ is the unit circle in the plane and $f:C\to C$ is a map not homotopic to the identity, then $f(x)=-x$ for some $x\in C$.

This from "Basic Topology" by Armstrong. I can't figure out what $f(x)=-x$ is doing when mapping from unit circles. Is this the antipodal map?
0
votes
1answer
43 views

Reference (or proof) of: classifying map $u\colon M \to K(\pi,1)$ induces $H^1(\pi_1(M);\mathbb{Z}/2)\cong H^1(M; \mathbb{Z}/2)$

I'm try to understanding a survey chapter in Angeloni - Metzler -Sieradski "Two dimensional Homotopy and combinatorial group theory" namely the one about Stable classification of $4$-manifold (see ...
1
vote
1answer
100 views

What's the cohomology of disjoint union of two circles

I am computing the cohomology of $T^2$ by Meyer-Vietoris sequence. $T^2$ can be seen as the union of two open sets U and V s.t. U and V are diffeomorphic to a cylinder respectively. Thus U$\cap$V is a ...
1
vote
0answers
38 views

Verify proof that $f:X\to Y$ a homeomorphism implies $f_*:H_p(X)\to H_p(Y)$ is an isomorphism

I have to prove that $f:X\to Y$ a homeomorphism implies $f_*:H_p(X)\to H_p(Y)$ is an isomorphism for all $p$ Where $H_p(X)$ is the $p$th homology group of $X$. To me this seems to come down to ...
3
votes
1answer
34 views

Is the base of a disc bundle necessarily a strong deformation retract of the total space?

I am reading Algebraic Topology by E.H.Spanier and in the proof of the Thom-Gysin map for disc bundles (on page 260) he says that $p : E \to B $ is a deformation retraction. I do not understand how ...
2
votes
0answers
34 views

Show that $k$-colorings of a link are in bijection with homomorphism $\pi_1(\mathbb{R}^3\setminus L)\to D_k$

Here $D_k$ is the group of symmetries of a regular $k$-gon. $D_k$ has $2k$ elements, the $k$ rotations through multiples of $2\pi/k$ and the $k$ reflections. I think this is related to Wirtinger ...
17
votes
1answer
463 views

Lecture Notes for Hatcher's Algebraic Topology

Hatcher's book Algebraic Topology is a standard text in the subject, and I was wondering if there were any lecture notes or even syllabi to accompany it. I am mostly concerned with sequencing, meaning ...
2
votes
0answers
69 views

3 sheeted cover of Klein bottle with torus

So I'm dealing with this exercise in which it is asked to determine whether the torus can be a 3-cover of the Klein bottle. A friend of mine came up with a proof that this is not the case, but this ...
2
votes
0answers
79 views

Uniqueness of homology functor from pairs of polyhedra

In my book it is written the following theorem Suppose that $H$ and $H'$ are exact homotopic functors from the category of pairs of polyhedra to the category of sequences of Abelian groups that ...
1
vote
0answers
57 views

The cohomology of the fiber of the Hopf fibration using the Eilenberg-Moore Spectral Sequence

In John McCleary's book about Spectral Sequences he computes on p.248 the cohomology of the fiber of the Hopf fibration $S^3 \to S^7 \to S^4$ using the Eilenberg-Moore Spectral Sequence. He deduces ...
1
vote
1answer
163 views

Why does there exist a continuous map with no fixed point $f\colon S^n\to S^n$ when $n\ge 1$?

Why does there exist a continuous map with no fixed point $f\colon S^n\to S^n$ when $n\ge 1$? I can find a continuous map that has no fixed points for the case $n=1$ but I fail to see how this ...
1
vote
1answer
26 views

Relative $0$-homology of $(\Delta^n, \partial \Delta^n)$

Let $\Delta^n$ be the standard $n$-simplex, with $n>0$. Denote with $H_0$ the (simplicial) $0$-homology. In my book it is written that $H_0(\Delta^n, \partial \Delta^n)=\mathbb{Z}$. But $\Delta^n$...
6
votes
0answers
116 views

Does a group homomorphism up to homotopy induce a map between classifying spaces?

Let $H$ and $G$ be topological groups and denote by $BH$ and $BG$ their classifying spaces. If $$f\colon H\rightarrow G$$ is a continuous group homomorphism, we get an induced map of spaces $$Bf\...
1
vote
1answer
43 views

Action of continuous function on boundary.

Let $T_r=\{x\in R^n:||x||<r\}$,$f:\overline T_r \rightarrow R^n$ is continuous condition 1: $\forall \lambda >0,x\in\partial T_r$ , $f(-x)\ne\lambda f(x)$ condition 2: $\forall \lambda >0,...
2
votes
1answer
33 views

Of what is the Hopf map the boundary?

Consider a generator $x$ of the singular homology group $H_3(S^3)$. I think of this (perhaps wrongly?) as something like the identity on $S^3$, cut up into simplices. Now we have the Hopf fibration $\...
0
votes
1answer
83 views

Homotopy commutative diagrams and homotopy equivalent spaces

The question is fairly general. Suppose I have a homotopy commutative diagram of the form \begin{equation} \require{AMScd} \begin{CD} A @>{f}>> B\\ @V{h}VV @V{i}VV \\ C @>{g}>> D \...
0
votes
1answer
30 views

The projection onto the orbit space $X/G$

Let $X$ be a locally compact, Hausdorff, path connected and locally path connected space. Assume a group $G$ acts freely and properly discontinuously on $X$, which means $\forall K^{compact},~~\{g\in ...
2
votes
1answer
61 views

The complement of a finite union of rectangles has only finitely many components

Problem. Assume that $V_i\subset \mathbb R^2$, $i=1,\ldots,n$, are open rectangles, and their sides are parallel to the axes. Show that $\mathbb R^2\setminus \bigcup_{i=1}^n V_i$ possesses finitely ...
11
votes
1answer
154 views

Is the homology class of a compact complex submanifold non-trivial?

Let $X$ be a connected complex manifold (not necessarily compact). Let $C \subset X$ be a compact complex $k$-dimensional submanifold (for some $k>0$). Is it true, in this generality, that the ...
4
votes
0answers
59 views

Curves Knotted in the Torus

I've been struggling with the following problem. I suspect it's actually not so hard, but I'm missing something relatively obvious about the proof. The attached image will help. Suppose $K$ and $L$ ...
3
votes
3answers
85 views

Two term free resolution of an abelian group.

This is probably a very easy question but I think I am missing some background regarding free abelian groups to answer it for myself. In Hatcher's Algebraic Topology, the idea of a free resolution is ...
0
votes
1answer
14 views

(Local path) Connectivity of the graph of the complex square root

I am wondering if the set $S := \{(z,w): w^2 = z, w \ne 0 \} \subseteq \mathbb{C}^2$ is a connected or locally path connected space under the subspace topology? This set lives in four dimensional ...
1
vote
1answer
44 views

Retracting one contractible space to another

What is an example of topological spaces $X \subseteq Y$ such that $X$ is closed in $Y$, and $X$ and $Y$ are both contractible, yet $Y$ does not retract to $X$? I'm having a hard time coming up with a ...
2
votes
1answer
124 views

The homology of wedge sum

This is an exercise of Bredon (pg. 190) which I tried to do but got stuck at one part. He asks the following: Let $X$ be a Hausdorff space and let $x_0 \in X$ be a point having a closed ...
3
votes
2answers
92 views

One point union, second homotopy group is not finitely generated?

Let $X$ be the one-point (wedge sum) union of the circle $S^1$ and the sphere $S^2$. What is the easiest way to see that the abelian group $\pi_2(X)$ is not finitely generated?
0
votes
1answer
57 views

$S^2-C$ is locally path connected.

Show that $S^2-C$, where $C$ is a simple closed curve, is locally path connected. I am reading the proof of Jordan Separation Theorem in Munkres book. At the very beginning, It states that "Because $...
1
vote
1answer
82 views

Induced Isomorphism on 2nd Homology Group

I have two related questions: 1) Is there a continuous map from the 2-sphere to the torus inducing an isomorphism on the 2nd (integral) homology groups? 2) Is there a continuous map from ...
1
vote
2answers
32 views

Homeomorphisms between two punctured planes

Question: Let $p,q$ be two different points in the interior of $D\subset\Bbb{R}^2$ where $D$ is the closed unit disk. Is there a homeomorphism $h:\Bbb{R}^2\setminus\{p\}\to\Bbb{R}^2\setminus\{q\}$ ...
0
votes
1answer
29 views

Covering group $Aut(\tilde{X},p)\cong NH/H $

I want to show $Aut(\tilde{X},p)\cong NH/H $ where $H=p_*\pi_1(\tilde{X},\tilde{x_0})$ and $NH$ is the normalizer of $H$. In my text book, the author sketches the proof of the above by using the ...
1
vote
1answer
94 views

Upper hemisphere of $S^2$ homeomorphic to the ball $B^2$

How can I show that the upper hemisphere E is homeomorphic to the ball $B^2$ I can see it intuitively, but I don't know how to prove it rigorously. I need it to prove the following theorem There is ...
5
votes
1answer
75 views

Continuous map with $f(x)\in B(x,r)$ for every $x\in\Bbb{R}^2$ with some fixed $r>0$ is surjective

Let $f:\Bbb{R}^2\to\Bbb{R}^2$ be a continuous map such that there exists $r>0$ with $f(x)\in B(x,r)$ for every $x\in\Bbb{R}^2$. Show that $f$ is surjective. I have some vague idea that if $p\...
4
votes
1answer
85 views

Fundamental group of real projective plane minus one point

I understand that the $\mathbb{R}P^2$ is homeomorphic to the unit disc with boundary points identified with their antipodes. But even if we puncture the disc and stretch it from the origin to let it ...
2
votes
1answer
40 views

Meaning of torsion elements in fundamental groups

I've been thinking today about the fundamental group of the projective plane which is the cyclic group of two elements ($\pi_1(\mathbb{R}P^2)=\mathbb{Z}_2$). This means there is only one class of ...