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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0
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1answer
11 views

Isomorphic homotopy groups of universal cover?

In Ralph Cohen's notes on the topology of fiber bundle he says (1) on pp.167, $BSO(n) \to BO(n)$ is a universal cover thus $\pi_i(BSO(n)) \to \pi_i(BO(n))$ is an isomorphism for $i \geq 2$ (2) on pp....
1
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0answers
17 views

Why is topological K-theory equivalent to nonabelian cohomology with respect to the stable unitary group?

I was reading on the $n$Lab page for topological K-theory that taking cohomology of a smooth space with respect to the smooth $\infty$-stack $\mathbf{Vect}$ is equivalent to taking its cohomology with ...
1
vote
2answers
27 views

category-theory, right group action

Let $G$ be a group. We observe the category $(Set)_G$ of right group actions. a) Let $X\times G\to X$ and $Y\times G\to Y$ be two transitive right group actions with $x\in X$ and $y\in Y$. Find a ...
2
votes
1answer
393 views

Deck transformation acting properly discontinuously assumed covering space is path-connected

Let $p:E\rightarrow X$ be a covering space, $x\in X$ fixed point, $E$ path-connected and $\Delta(p)$ – Deck transformation group of $p$, that is $\Delta(p) = \{f\in \text{Homeo}(E):pf=p\}$. Let $\pi_1(...
1
vote
1answer
33 views

tom Diecks's proof of $H_1(X)\cong \pi_1(X,x_0)^{ab}$

My question is about tom Dieck's proof of Theorem 9.2.1 on page 227, which states that if $X$ is path connected, then the induced map $$h:\pi_1(X,x_0)^{ab}\to H_1(X)$$ is an isomorphism. Specifically,...
1
vote
1answer
55 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 ...
0
votes
0answers
11 views

Duality between Thom space and a manifold embedded into a sphere

In a document https://www.math.purdue.edu/~gottlieb/Bibliography/53.pdf (s. 19) it is mentioned that there is a map $S^n \to M^+ \wedge Th(\nu (M, S^n))$, which gives a Spanier-Whitehead duality ...
6
votes
2answers
75 views

Infinite sequence of distinct spaces, all with same homology

Using the following fact, we get infinitely many non-homotopic maps $f_k:S^{2n-1}\to S^n\vee S^n$. Fact: $\pi_{2n-1}(S^n\vee S^n)$ contains a $\Bbb Z$-summand. So we can consider the spaces $X_k=...
1
vote
2answers
35 views

How to construct a homotopy equivalence between a mobius band and a circle?

A mobius band is homotopic equivalent to a circle because the mobius band can deformation retract onto a circle. I am wondering how could we understand this fact from the definition of being ...
0
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0answers
23 views

Proof of Brouwer fixed point theorem using change of variable formula. [on hold]

Is there a proof of Brouwer fixed point theorem using change of variable formula for integration?
4
votes
1answer
38 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(...
8
votes
2answers
146 views

What is the class of topological spaces $X$ such that the functors $\times X:\mathbf{Top}\to\mathbf{Top}$ have right adjoints?

For any topological space $X$, define a functor $\times X:\mathbf{Top}\to\mathbf{Top}$ by $Y\mapsto Y\times X$ (and acting on the hom-sets in the natural way). I know that if $X$ is locally compact, ...
0
votes
2answers
46 views

To show that product $Z=X×Y$ in the product topology is a CW complex

I would like to prove the following: If $X,Y$ are CW complexes, and either $X$ or $Y$ is locally compact then the product $Z=X×Y$ in the product topology is a CW complex. (-see here) In order to ...
10
votes
3answers
1k views

Is homology with coefficients in a field isomorphic to cohomology?

is it true that when we compute homologies and cohomologies with coefficients in a field then homology and cohomology groups are isomorphic to each other? That is valid when homology groups are free ...
4
votes
1answer
75 views

Graph theory application of homology

I am struggling with the idea of local homology groups and would like to see an example of how to go about finding them in general. I'm thinking of the most trivial case to apply the theory of local ...
0
votes
0answers
14 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!
0
votes
1answer
48 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 ...
3
votes
0answers
46 views

Obstruction for extending a map along a CW-inclusion: sufficient conditions?

I'm looking for a clarification of the highlighted comment taken from "A User's Guide to Algebraic Topology" by Dodson & Parker. In order to make the setting clear, I uploaded definition $8.1.6$ ...
1
vote
1answer
16 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: ...
1
vote
1answer
50 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 {...
5
votes
1answer
79 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 $ ...
0
votes
1answer
37 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\...
0
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0answers
42 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 ...
-1
votes
1answer
49 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 ...
0
votes
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 ...
0
votes
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 ...
1
vote
1answer
32 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))\...
0
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0answers
25 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
vote
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 ...
0
votes
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) ...
0
votes
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 ...
3
votes
2answers
65 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
votes
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 ...
1
vote
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 ...
8
votes
1answer
112 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 ...
2
votes
1answer
33 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 ...
2
votes
1answer
45 views

Why does $f(x,z)=(x,z^2/|z|)$ have degree $2$?

Write the $n$-sphere as the set $S^n\approx \{(x,z)\in\Bbb R^{n-1}\times\Bbb C: |x|^2+|z|^2=1\}$, and define a mapping $f: S^n\to S^n$ by $f((x,z)) = (x,\frac{z^2}{|z|})$. Why is $\deg(f)=2$? ...
15
votes
3answers
3k views

An intuitive idea about fundamental group of $\mathbb{RP}^2$

Someone can explain me with an example, what is the meaning of $\pi(\mathbb{RP}^2,x_0) \cong \mathbb{Z}_2$? We consider the real projective plane as a quotient of the disk. I don't receive and ...
3
votes
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 $B^4$?
1
vote
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
votes
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 ...
0
votes
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
votes
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....
0
votes
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 ...
5
votes
1answer
441 views

If two maps induce the same homomorphism on the fundamental group, then they are homotopic

This is exercise 15.11(d) in C. Kosniowsky book A first course in algebraic topology: Prove that two continuous mappings $\varphi,\ \psi:X\to Y$, with $\varphi(x_0)=\psi(x_0)$ for some point $x_0\...
0
votes
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 ...
3
votes
0answers
102 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 ...
3
votes
1answer
66 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. ...
0
votes
1answer
65 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
votes
2answers
160 views

$\pi_n(X^n)$ free Abelian?

I have encountered a problem which states that denote $X$ as an Eilenberg-MacLane space $K(G,1)$ and is a CW complex, show that $\pi_n(X^n)$ is free Abelian for $n \geqslant 2$. However, I think I ...