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
11 views

homology theory for $C^*$-algebras, map is natural wrt morphisms of short ecact sequences

I want to assure me if I understand the part of the exactness axiom that "$\delta$ is natural" corretly and if not, then my question is: what does it mean? The setting is the following definition: ...
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1answer
43 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
36 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
36 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
16 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 ...
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1answer
46 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
72 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
40 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
44 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
24 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). ...
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1answer
31 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
48 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
12 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
27 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
39 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
63 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
41 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 ...
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1answer
40 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
31 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 ...
3
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1answer
55 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
17 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
37 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....
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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
111 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
64 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
64 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
55 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
69 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
41 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 $g_{0} \circ f_{0} \simeq g_{1} \circ f_{1}$ ...
4
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0answers
69 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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0answers
98 views
+50

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
150 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
39 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 ...
1
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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 ...
1
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0answers
39 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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0answers
32 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(...