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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1answer
53 views

A proposition about free product

My question is about a proposition that I found on Munkres book: Topology (page 419). My definitions are based on this book. I give you the definition of free product that he uses. $\textbf {...
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0answers
44 views

$H_{k+1}(X \cup_f D^{k+1},X) = ?$

I am stuck with the calculation of the following homology group: $H_{k+1}(X \cup_f D^{k+1},X) = ?$ where $X$ is a simply-connected CW complex and $f: S^k \to X$ is a continuous map (attaching map of ...
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1answer
41 views

CW complex= cell complex?; compact metrizable spaces

I have a question about finite cell complexes and compact metrizable spaces. In a paper I read the statement: Let $X$ be a compact metrizable space. Then $X$ is a countable inverse limit $\varprojlim\...
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1answer
19 views

Seifert matrix, linking numbers, generators

I have been asked to compute the seifert form of a knot, the twist knot. I know how to compute the seifert surface, and then the seifert matrix seems to be defined accordingly (according to all the ...
-1
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1answer
56 views

Which group homomorphisms induce the action of the fundamental group on the fiber?

Given topological spaces $X$ and $Y$ and a covering map $p: X \rightarrow Y$, we know that the group $\pi_1(Y,y_0)$, where $y_0\in Y$, acts on the fiber $F=p^{-1}(y_0)$. Also, we know that the set ...
5
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1answer
93 views

$[X,Y]$ is finite where $X$ is finite connected CW-complex, and $Y$ has finite homotopy groups

I have read this question in Allen Hatcher's book Algebraic Topology, (exercise 20, page 359): Show that $[X,Y]$ is finite if $X$ is a finite connected CW complex and $ \pi_i(Y) $ is finite for $ ...
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1answer
42 views

Homology group of an open set on $S^1$

Let $U$ be an open set which is constructed as intersection of $S^1$ and open ball in $\mathbb{R}^2$. And $x$ is just a point contained in $U$. My opinion: By long exact sequence, $H_n(U, U-x)$ is ...
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1answer
50 views

A relation between the index of a fundamental group and the covering map

My professor just enunciated this statement: $|\pi_1(X,x_0):\pi_1(\tilde X,\tilde x_0)|=\#P^{-1}(x_0)$ where $P$ is the covering $P:\tilde X\rightarrow X$, such that $P(\tilde x_0)=x_0$. I've ...
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0answers
27 views

Differences among the Group Cohomology with coefficients over a commutative ring and coefficients over an arbitrary $G$-module

Assume that $G$ is a finite group and $k$ is an arbitrary commutative ring. From the general theory we know that the group cohomology $H^{*}(G ; k)$ becomes a graded ring (with the cup product). ...
1
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1answer
33 views

Deck transformation, covering space

Let be $X=(S^1-1)\cup (S^1+1)\subset\mathbb{C}$ (shaped like the "eight") and $u(t)=e^{2\pi it}-1, v(t)=1-e^{2\pi it}$. Give every deck transformation $\Delta(p)$ and $p_{\ast}(\pi_1(Y, y_0))\...
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0answers
50 views

Is the cohomology ring of a CW complex computable?

There is a well-developed technology for computing the cohomology groups of a CW complex, cellular cohomology. It reduces the problem of computing cohomology to the two simpler problems of (1) ...
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1answer
28 views

Is there a common way, to find all deck transformations $\Delta(p)$

Suppose $p:E\longrightarrow B$ is a covering projection. I have a general question, on how to find the group of all deck transformations $\Delta(p)$. Is there a common way to do this, or what could be ...
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0answers
15 views

Covering spaces of surface sphere glued to the mobius strip at one point on its boundary.

I have determined the universal covering space, but I am having trouble finding two-sheeted and three sheeted covering spaces. Any help would be greatly appreciated on how to approach this!
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0answers
28 views

definition of mod p k-theory

The (topological) complex K-theory is a cohomology theory, i.e can be represented by a spectrum $K$ whose $2n$-th space is $BU \times \mathbb{Z}$ and whose $2n+1$-th space is its loop space (and is ...
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0answers
41 views

Projective space, fundamental group

Let $g:\mathbb{R}P^2\to\mathbb{R}P^2$ continuous. Let $q:S^2\to\mathbb{R}P^2$ die quotient map. Show, that: If $g_{\ast}(\pi_1(\mathbb{R}P^2, x))$ is not trivial (therefore contains more ...
3
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2answers
69 views

Why every map $f : S^n \to T^n (n>1)$ has topological degree zero?

Why every map $f : S^n \to T^n (n>1)$ has topological degree zero? I don't know anything about covering spaces, and has been told to me that this assertion comes from this theory! I do appreciate ...
3
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0answers
43 views

Let $\gamma_1, \gamma_2 : I \to M$ two homotopic curves, $M$ a $C^{\infty}$ manifold. If $\omega$ is a closed $1-$form then:

Let $\gamma_1, \gamma_2 : I \to M$ two homotopic curves, $M$ a $C^{\infty}$ manifold. If $\omega$ is a closed $1-$form then: $$\int_{\gamma_1(I)} \omega = \int_{\gamma_2(I)}\omega.$$ Then, conclude ...
1
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1answer
69 views

Troubles to understanding notation and some terminology on cohomology of finite groups

I am reading Adem, Milgram's book "Cohomology of finite groups" and I have some troubles with the notation. In particular, I don't get whether they are referring to the cohomology as a ring or as a ...
2
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1answer
34 views

Correlation amomg cohomology of a group $G$ and singular coohomology of its classifying space $BG$

Assume that $G$ is a finite group and let's denote by $BG$ its classifying space. Then we know that we can construct the cohomology $H^{*}(G ; M)$ with (local) coefficients, for a $G$-module $M$, in ...
4
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1answer
61 views

Homology 4-balls with boundary $S^3$

Are there interesting homology 4-balls with boundary $S^3$? Going the other way, must any homology 4-ball with boundary $S^3$ be homotopy equivalent/homeomorphic/diffeomorphic to $B^4$?
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1answer
18 views

How do I see if the induced homomorphism from the inclusion map $S^{1'}\to S^1\times S^3$ is injective

Let $S^1\times S^3$ be parametrised as $\{(\alpha,\beta, \gamma)\in \mathbb{C}^3||\alpha|^2+|\beta|^2=1, |\gamma|=1\}$ and let $S^{1'}=\{e^{i\theta}(1,0,1)|\theta\in [0,2\pi]\}$. I would like to see ...
0
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1answer
36 views

projective space, quotient map, existence of a function

Let $g:\mathbb{R}P^2\to\mathbb{R}P^2$ continuous. Let $q:S^2\to\mathbb{R}P^2$ denote the quotient map. Show: It exists a function $h: S^2\to S^2$ with $q\circ h=g\circ q$ Hello, I really ...
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0answers
27 views

covering space, deck transformation

Let $X=(S^1-1)(S^1+1)$ be the eight, $Y=\mathbb{R}\cup\{z+2k\pi+i:k\in\mathbb{Z},z\in S^1\}\subset\mathbb{C}$ and $u(t)=1-e^{2\pi it}, v(t)=e^{2\pi it}+1$ a) Give (per formula) a covering space $...
3
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2answers
39 views

The Incluson Map $S_1\to S_1\times S_1$ Induces an Injection in First Homology.

Let $T=S^1\times S^1$ be the torus and $A=S^1\times \{x_0\}$ be the "vertical" circle in the usual depiction of the torus as a tyre tube sitting "horizontally". Let $i:A\to T$ be the inclusion map....
0
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1answer
18 views

Proving weak homotopy equivalence of a map

This is a modification of the question I previously asked here. Consider the category of pointed topological spaces $C$. Suppose objects $a,b,c,d,e,f,g,h \in C$. Suppose we also have the commuting ...
8
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1answer
118 views

What is this manifold?

As picture below ,it is a Mobius band with a cylinder crossing it .Let it be $\Omega$ . Obviously , $\partial \Omega$ is a circle. Now , what is $\Omega/\partial \Omega$ ( I mean glue the boundary to ...
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0answers
71 views

Homology Groups of Tangent 2-Spheres

I have been trying to compute the Homology Groups $ H_n $ of two tangent 2-Spheres (we will call this space, X). By previous results, I am able to easily determine that $ H_0(X) $ is isomorphic to the ...
0
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1answer
70 views

Cayley graph of the fundamental group of the 2-torus

Does anybody knows some description or some good picture on the internet for the Cayley graph of the fundamental group of the 2-torus, when this is constructed by connected sum of two torus. I know, ...
5
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1answer
71 views

The Euler Characteristic of $\mathbf RP^2$ is a Fraction.

Problem 22 in Section 2.2 in Hatcher's Algebraic Topology reads For $X$ a finite CW complex and $p:\tilde X\to X$ an $n$-sheeted covering space, show that $\chi(\tilde X)=n\chi(X)$. Here $\chi$ ...
3
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1answer
68 views

Non contractible space with contractible suspension

I know that there exist non-contractible spaces $X \not\simeq \ast$ with contractible suspension $\Sigma X \simeq \ast$. For instance the 2-skeleton of the Poincaré homology 3-sphere is such a space. ...
3
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1answer
60 views

Correspondence $\{$principal $G$-bundles on $M\}\leftrightarrow\{$conjugacy classes of homomorphisms $\pi_1(M)\to G\}$

Context. I'm reading Qiaochu's short note Surfaces and the representation theory of finite groups which aims to prove Mednykh's formula inspired by ideas from topological quantum field theory. On page ...
2
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2answers
72 views

In the quotient topology $D^2/{S^1} \cong S^2 $

Let $X = D^2 = \{(x, y) \in R^2\ : x^2 + y^2 \le 1\}$ be the closed unit disc (in the standard topology). Identify $S^1$ with the boundary of $D^2$. Now I have to prove that $$D^2/S^1\cong S^2$$ I ...
3
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2answers
70 views

Is there any 'nice' space with fundamental group $\mathbb{Z}_3$?

I'm trying to build up intuition for the fundamental group, as it occurs in physics. In the simplest examples, the fundamental group is trivial, or $\mathbb{Z}$, or $\mathbb{Z}^n$. We can also get $\...
1
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1answer
47 views

Arc connectedness of Telophase topology

In Counterexamples in Topology Book by Lynn Steen i found that Telophase topology is arc connected. How can we build an arc in this topology ? And also could you give me an idea of how to prove that ...
2
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2answers
32 views

Compositions of homotopic maps are homotopic

I'm reading some lecture notes on homotopy, and the author has just proved the theorem: If $f_{0} \simeq f_{1}$ and $g_{0} \simeq g_{1}$, then $g_{0} \circ f_{0} \simeq g_{1} \circ f_{1}$ ...
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0answers
71 views

what is a (co)homology theory?

There are different axioms for a (co)homology theory: Homotopy invariance is always there, but for the rest there is or isn't: long exact sequence for pairs of topological spaces exact sequence for ...
3
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2answers
153 views

projective space, constant function, homotopy

Show, that every continious function $f:\mathbb{R}P^2\to S^1$ is a homotopy to a constant function. (Hint: Has $f$ a lift $\tilde{f}:\mathbb{R}P^2\to\mathbb{R}$) Hello, I want to solve this ...
2
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1answer
95 views

Generators of the fundamental groups of the 8-figure and the torus

I have two doubts strictly related to each other. 1) Firstly, consider the $8$-figure, namely the union of two circles in a point $x_1$. Using the Seifert-Van Kampen's theorem I proved that its ...
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0answers
94 views

“admissible” maps from context

I have been reading Massey's Algebraic Topology and on page 158 came across the following "semi-mystical principle" which he says guides much mathematical research: Whenever we wish to gain ...
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0answers
166 views

Deformation retract of wedge sum

Let $(X_\gamma)_{\gamma \in \Gamma}$ be a collection of topological spaces, and let $x_\gamma \in X_\gamma$ be a fixed point for each $\gamma$. Fix some $\alpha \in \Gamma$, and suppose that for $\...
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1answer
49 views

Characterizing spaces with no nontrivial covers

I know that simply connected locally path-connected spaces have no nontrivial covers. Is there a characterization of spaces with this property?
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2answers
45 views

Hatcher basic terminology/phrasing

I'm trying to self-study some algebraic topology, reading Hatcher. His questions seem much less straightforwardly worded than Munkres - with Munkres it was always clear that you weren't expected to ...
1
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1answer
39 views

Proof of More Flexible Mayer-Vietoris for Calculating Homology Groups

On pg. 150 of Hatcher's Algebraic Topology, the author writes: Let $X$ be a topological space and $A$ and $B$ be subspaces of $X$ such that $X=A\cup B$. Suppose there are open subspaces $U$ and $V$ ...
3
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1answer
36 views

Preimage of a simply closed curve under the two-dimensional antipodal map

Suppose $p:S^2\to P^2$ is the quotient antipodal map, and $J$ is a simply closed curve in $P^2$, then $p^{-1}(J)$ is either a simply closed curve in $S^2$, or two disjoint simply closed curves in $S^2$...
3
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1answer
40 views

Cohomology class of zero set of a section

Say we have a rank $r$ smooth vector bundle $E\to X$ and a smooth section $s:X\to E$ of it. Shortly after the 30 min mark in this video Joe Harris defines the $r$th Chern class of this bundle to be ...
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1answer
18 views

Relative Homology of the Mapping Cylinder w.r.t a Subspace

Given a continuous map $f:X\to Y$, the mapping cylinder of $f$ is defined as the space obtained from $(X\times I)\sqcup Y$ by identifying $(x, 1)$ with $f(x)$ for all $x$. Let $f:S^n\to S^n$ be a ...
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0answers
41 views

Bott's topological obstruction to integrability

Let $M$ be a smooth manifold, $E\subseteq TM$ be an integrable distribution and $\pi:TM\to Q=\frac{TM}{E}$ the quotient bundle with $\dim Q=q$. Bott showed that \begin{equation} Pont^k(Q)=0 \qquad \...
4
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1answer
43 views

Concluding $\Bbb Z$-cohomology from $\Bbb Z_2$-cohomology using Bocksteins

According to a theorem of Serre, the cohomology algebra $H^*(K(\Bbb Z,3); \Bbb Z_2)$ is a polynomial ring on elements $\iota_3, \,\operatorname{Sq}^2(\iota_3), \,\operatorname{Sq}^4\operatorname{Sq}^2(...
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0answers
28 views

Proof of the cellular boundary formula

I'm trying to understand the proof in Hatcher (p. 141) of the cellular boundary formula. Now there's one thing that Hatcher does several times in his book and that I don't understand very well: he ...
1
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0answers
57 views

Proving a function isn't homotopic to a map to the boundary

Let $X$ be a compact Hausdorff space and $S$ a finite dimensional, convex, Hausdorff space. Moreover, let $f:X \to S$ be a closed, surjective map with $\tilde{H}^q(f^{-1}(s))=0$ for all $q\ge 0$ and $...