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

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3
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1answer
68 views

Fundamental groupoid of a contractible space

I read that the fundamental groupoid of a contractible space is indiscrete. How can one show this? I found this as an exercise here.
2
votes
0answers
15 views

Elementary Cube vs. Elementary Chain

I am reading Computational Homology by Kaczynski, Mischaikow, and Mrozek. On page 47, for every elementary cube, $Q \in \mathcal{K}_k^d$ they associate an object $\widehat{Q}$ that they call an ...
1
vote
1answer
38 views

The fundamental group of a plane without a finite number of points.

How can I calculate de fundamenta group of $\mathbb{R}^2$ without a finite number of points? I know that the answer should be $S^1\vee S^1 \vee \cdots \vee S^1 $ where the product repeats as many ...
2
votes
0answers
41 views

Deck transformation

I read that a deck transformation is uniquely defined by the value of one point. Unfortunately, I don't understand where this comes from. I mean, all we know is that there is one point in the fibre ...
1
vote
1answer
42 views

What may be the number of connected components of this randomly generated topological space?

I hope there exists some answer to this but I am not very good in probability so i do not know how to tackle this problem. It's contemporary-art related so it might not have a good answer. ...
4
votes
1answer
72 views

Fundamental polygon square $abab$

What is the most convenient description of the space with fundamental polygon a square, with all vertices identified, glued by $abab$? If we were to identify only opposite vertices, we would get ...
0
votes
0answers
26 views

Winding number of $S^1$ vector fields with $|u| > |v|$

Let $u$ and $v$ are nonvanishing vector fields on $\mathbb{S}^1$ and $|u(z)| > |v(z)|$ at every point of $\mathbb{S}^1$. Prove that $deg(u) = deg(u + v)$. My idea is to take a homotopy $h_t(z) = ...
1
vote
1answer
62 views

Properties of Pushout

suppose we have a pushout square in $\mathrm{Top}$: \begin{align*} \require{AMScd} \begin{CD} X_0 @>{\mu_1}>> X_1\\ @V{\mu_2}VV @VV{\alpha_1}V \\ X_2 @>>{\alpha_2}> X ...
1
vote
1answer
48 views

Homeomorphy of a surface

I am studying graphs on surfaces (i.e. maps). Their definition is below: We call map a representation $(X,\mathcal{D})$ of a finite connected graph $\Gamma=(V,E)$ in the topological surface $X$ ...
4
votes
1answer
49 views

Does there exist a double cover with trivial deck transformation group?

Sorry for the naive question. The following statement at the beginning of Bredon, chapter 4, §20, got me confused: Let $\pi:X \to Y$ be a two-sheeted covering map. Let $g:X \to X$ be the unique ...
4
votes
1answer
130 views

Can Somebody Please Outline a Reading Course For Me in Algebraic Topology

I want to start self studying algebraic topology and I am looking for guidance regarding the same. In the past I have made the mistake of trying to learn a mathematical subject by reading fat books ...
0
votes
2answers
66 views

Homotopy/fundamental group question: Why group axioms fail when defined on paths?

Neither Munkres nor Lee in their textbooks explicitly show why (fundamental) group properties like associativity fail when defined at the level of paths but work fine for homotopy classes of paths. ...
2
votes
0answers
42 views

Calculating the fundamental group of the Klein Bottle using the Seifert-Van Kampen theorem

I want to calculate by two different way the fundamental group of the Klein Bottle. First one: I want to use that the Klein Bottle is can be decomposed in two Mobiüs Band as the following picture ...
2
votes
1answer
47 views

Left and right action?!

The adjoint of the adjoint representation $Ad^* : G \times \mathfrak{g}^* \rightarrow \mathfrak{g}^*, (g,x) \mapsto Ad^*_{g}(x)$ is a group action on the dual space of the Lie algebra. Now, we said ...
1
vote
1answer
30 views

What is a linear embedding from a simplex $\Delta^n \to \mathbb{R}^n$?

As stated in the title, reading Milnor-Stasheff Characteristic classes, I encountered at page 95 the following sentence: let $\Delta^n$ be an $n$-simplex, linearly embedded in the $n$-dim vector ...
5
votes
0answers
76 views

Twisted nail puzzle framed in terms of algebraic topology?

See here for a description of the puzzle: Twisted Nail Puzzle. My question is, can someone provide a description of the puzzle and its solution in context of the language of algebraic topology?
2
votes
1answer
48 views

question about “Homology fibrations and the group completion theorem”

In the paper Homology fibrations and group completion theorem, McDuff-Segal, page 281 line 17-line 18: we have a fibre bundle $M_\infty\to (M_\infty)_M\to BM$ with $(M_\infty)_M$ constractible. In ...
0
votes
0answers
30 views

Compute the singular homology group of a “rational optical grating”

Let $X$ be the subspace of the square $I \times I$ consisting of the four boundary edges plus all points in the interior whose first coordinate is rational. Calculate the singular homology ...
1
vote
0answers
29 views

Lifting of certain $S^1$ valued maps

It's well-known that every path $s(t): I \to S^1$ has a lifting, i.e. mapping $\widetilde{s(t)}: I \to \mathbb{R}$, so that $e^{i\widetilde{s(t)}} = s(t)$. The main idea of constructing ...
0
votes
0answers
56 views

Calculate the fundamental group of $S^1/\mathbb Z_n$

Calculate the fundamental group of $S^1/\mathbb Z_n$ ,where $\mathbb Z_n$ acts naturally on $S^1$ by rotations of $2\pi /n$ The origin of this problem is the following unclear solution of another ...
1
vote
0answers
53 views

homology of mapping telescope of a monoid

Let $M$ be a monoid with multiplication $\cdot$, $\pi_0(M)=\mathbb{N}$, and $m\in M$ in the component $1\in \mathbb{N}$ . We form a mapping telescope $$ M\overset{ {\cdot m}}\longrightarrow M\overset{ ...
2
votes
1answer
89 views

A map $h:S^1\to X$ Induces a Trivial Homomorphism of Fundamental Groups Iff it is Nullhomotopic.

I recently started reading Algebraic Topology from Part II of Munkres' book Topology(Second Edition). A part of Lemma 55.3 in the book proves the following: Let $h:S^1\to X$ be a continuous ...
0
votes
2answers
77 views

Lists of the first fundamental group of spaces. [closed]

Here are some list to start with $$\begin{array}{c|c|c|} \hline Space(S)& \pi_1(S) \\ \hline \mathbb{R}^2&0 \\ \hline \mathbb{S}^1& \mathbb{Z} \\ \hline 1-Torus& ...
1
vote
1answer
74 views

The winding number (topological definition) is well-defined

can someone please tell me if my proof of the lemma below is legit. At first I like to give a definition so you know what the lemma is about. Definition: Let $g:[0,1] \to S^{1}$ be a closed path in ...
1
vote
0answers
54 views

Computing the cohomology of the pair $(S^n\times S^n,D)$

Let $D=(x,x)\subset S^n\times S^n$ be the diagonal, and assume $n$ is even. I need to prove that the following sequence (taken from l.e.s of the pair) is exact $$0 \rightarrow H^n(S^n\times ...
1
vote
0answers
38 views

configuration space model for classifying space of monoid

Let $M$ be a monoid and $BM$ be its classifying space. There is a model for $BM$ based on labelled configuration spaces of the line $[0,1]$. Points of the configurations are labelled by elements of ...
2
votes
2answers
76 views

Relationship between homology of suspension of $X$ and $X$

The exercise is the following: Show that, for any homology theory (satisfying the usual axioms), there is a natural isomorphism $ \tilde{H_i}(X) \rightarrow \tilde{H}_{i+1}(\Sigma X)$. Well, I ...
0
votes
1answer
48 views

Representations of Fundamental Group and Monodromy

I have two representations of the fundamental group and I am under the impression they are the same. Any help in seeing this would be great. Preliminaries: Let $\phi: E \to M$ be a n-fold covering ...
0
votes
0answers
18 views

Locally ringed space locally isomorphic to a *closed* subset of $\mathbb{R}^n$

To me it's more natural to think of e.g. a tetrahedron as a closed subset of $\mathbb{R}^3$ than as a "manifold with corners" in the traditional sense -- i.e., locally isomorphic to open subsets of ...
6
votes
2answers
72 views

Number of roots the degree of the map?

Let $p$ be a polynomial function on $\mathbb{C}$ which has no root on $S^1$. My question is as follows: does the number of roots, up to multiplicity, of $p(z) = 0$ with $|z| < 1$ necessarily equal ...
2
votes
0answers
88 views

Fundamental group of the Klein bottle

Proof that fundamental group of the Klein bottle have the next two different presentations: a) $Gen(a,b \,; baba^{-1}=1)$, b) $Gen(a,b \,; a^{2}b^{2}=1)$. I have the proof of (a) and I can prove that ...
2
votes
1answer
58 views

Properly discontinuous action on manifold

I am actually not familiar with topology, but since we had a short outlook on these things in our differential geometry lecture today, I would appreciate some general remarks: Let $M$ be a smooth ...
1
vote
0answers
22 views

Categorical Notion of Quotient in Spectra

Having done some reading on spectra recently, I noticed that the definition of a quotient spectra for a closed subspectrum of a CW spectrum is simply given by taking the quotient of each of the spaces ...
8
votes
2answers
192 views

Can one prove that the fundamental group of the circle is $\mathbb Z$ without using covering spaces?

I am curious if there is a decent "bare hands" proof that the fundamental group of $S^1$ is $\mathbb Z$ that does not invoke covering space theory. One must show two claims. First, that $f(t)=e^{2\pi ...
2
votes
0answers
44 views

Different groups with the same classifying space.

Let $G$ be a topological group and $BG$ its classifying space. From the LES of the universal bundle, we get $\pi_i(BG)\cong\pi_{i-1}(G)$, so given the classifying space, we know all homotopy groups of ...
2
votes
1answer
37 views

A compact Lie group modulo by its maximal torus has nonzero Euler characteristic

In Andrew Baker's Matrix Groups, (in the proof of Theorem 20.11), there is an unproven statement that if $G$ is a compact Lie group and $T$ is a maximal torus, then $\chi (G/T)\ne 0$. I have an ...
5
votes
0answers
35 views

Homeomorphism between boundaries of open sets is “surjective”

Let $U,V\subset\mathbb{R}^n$ two bounded open sets and let $F:\overline{U}\to\mathbb{R}^n$ be a continuous map. Assume that $F$ maps $\partial U$ homeomorphically onto $\partial V$. The question: is ...
0
votes
2answers
113 views

Inclusion induces identity on homology

Let $(H_*, \partial_*)$ be a homology theory with values in the category of $\Bbb{Z}$-modules satisfying the dimension axiom. Then the inclusion $S^1\vee S^1\to T^2$ should induce (up to isomorphism) ...
0
votes
0answers
14 views

Local homology of a fibred product

Let $A,B$ be topological spaces and suppose that for $a\in A$ and $b\in B$ the local singular homology groups $H_k(A,A\setminus\{a\};\mathbb{Q})$, $H_k(B,B\setminus\{b\};\mathbb{Q})$ are known for all ...
0
votes
0answers
19 views

$ \pi : O(n) \rightarrow O(n)/O(n-k) \cong V_{n,k}(\mathbb{R}) $is a principal $ O(n-k) $-bundle.

I'm trying to prove that $ \pi : O(n) \rightarrow O(n)/O(n-k) \cong V_{n,k}(\mathbb{R}) $; $ A \longmapsto (Ae_1, ... ,Ae_k) $ (the projection from the orthogonal group to the Stiefel manifold) is a ...
1
vote
0answers
41 views

finite dimensional CW complex

Let $X$ be a finite dimensional CW complex, where X is simply connected and at least one $H_{i}(X)$, is non trivial (so that X not be contractible). Can we conclude that X has at least one non trivial ...
1
vote
0answers
50 views

Application of Morse Inequalities

I am an undergraduate student interested in morse theory. I understand, that the morse inequalities provide an lower bound for the number of critical points morse functions on a manifold can take. One ...
12
votes
2answers
133 views

Is $\mathbb{R}^n$ properly homotopy equivalent to $\mathbb{R}^m$ if $n \neq m$?

$\DeclareMathOperator{\id}{id} \newcommand{\R}{\mathbb{R}}$ If $f,g : X \to Y$ are two maps (all maps considered are continuous here), a homotopy between $f$ and $g$ is a map $H : [0,1] \times X \to ...
1
vote
1answer
36 views

what means 'the realization of a topological category'

In the paper Homology Fibrations and the "Group-Completion" Theorem. page 280 bottom line 10-bottom line 12, what means 'the realization of a topological category'?
1
vote
1answer
124 views

Fundamental group of $ \mathbb{S}^{n-1}\times\mathbb{R}$ minus $k$ disks $\mathbb{D}^n$

Let $X$ be the space obtained from $ \mathbb{S}^{n-1}\times\mathbb{R}$ by deleting $k$ disjoint subsets, each one homeomorphic to $D^n$. What is the foundamental group of $X$?
2
votes
1answer
41 views

Graph embedding into a surface

For example, let's consider a $K_{5}$ (complete graph on 5 vertixes) and a torus, which is defined as $S^{1} \times S^{1}$. How to build a continous embedding $f:K_{5} \rightarrow \mathbb{T}^{2}$? We ...
4
votes
2answers
67 views

If $G$ is a finite nontrivial group, then $K(G, 1)$ cannot be a finite CW-complex?

I am currently working on this algebraic topology problem and got stuck: Suppose $X$ is a finite CW complex with $\pi_1(X)$ a nontrivial finite group. Show that its universal cover $\widetilde{X}$ ...
8
votes
1answer
108 views

Does every continuous action of $S^1$ on $R^n$ have a fixed point?

I certainly can't think of one that doesn't. I am aware that there are decompositions of $R^n$ as a union of embedded $S^1$'s, but none of these seem like they would support a continuous action. ...
2
votes
1answer
43 views

Find a presentation for the fundamental group of $P^2\#T$

I have to find a presentation for the fundamental group of $ P^2\# T $. Which I believe to be identified by the following labeling scheme: $(a_1b_1a_1b_1)(a_2b_2a_2^{-1}b_2^{-1})$. $P^2$ is the ...
0
votes
1answer
19 views

Weak equivalence of ordinary and homotopy colimits

I am looking for conditions under which colimits and homotopy colimits of diagrams of, say, topological spaces, are weakly equivalent. I would appreciate answers not demanding an all too profound ...