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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32 views

A question regarding Hatcher's exercise problem [duplicate]

I'm studying Hatcher's algebraic topology, and stuck at a question. Here what I understand is that the question requires me to describe RP^n as a quotient of a delta complex and S^n. Is my ...
0
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2answers
57 views

Is the quotient map $X\to X/G$ a covering map?

Let $G$ be a group acting on a topological space $X$. Suppose that every $x,x' \in X$ that are not in the same orbit of the $G$-action have open neighborhood $U$ and $U'$ such tath $g(U)\cap U'=\phi$ ...
0
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0answers
54 views

$CP^n$ and $RP^n$, difference, simply connected, etc

In this question, i want to know the different properties of $CP^n$ and $RP^n$. If you know more about them please let me know. First, the definition of is different. $CP^n$ can be defined by $C−0$ ...
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0answers
28 views

showing that product of triangulable spcaes triangulable

the question is same as the title. I tried to decompose the product space into parts such that each is triangulable, but now am stuck. Could anyone help me how to prove the fact rigorously?
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0answers
45 views

Fundamental theorem of Galois covers

I attempted the proof in Tamas Szamuely's book, Galois groups and fundamental groups. I am stuck in the last portion. The cover $q \colon Z \to X$ is Galois if and only if $H$ is a normal subgroup of ...
2
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1answer
19 views

How to prove that a simplicial complex is path-connected if connected?

If K consists of finite simplices and connected, it seems intuitively clear that any two 0-simplices can be connected by a path whose image is a collection of 0-simplices and 1-simplices. But I can't ...
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2answers
50 views

How to prove that $S^{2n+1}/S^1$ is homeomorphic to $\mathbb CP^n$ under a given identification

We represent an element $(x_1,y_1,...,x_n,y_n,x_{n+1},y_{n+1})\in S^{2n+1}$ as an element $(z_1,...,z_{n+1})\in \mathbb C^{n+1}$ where $z_k = x_k+iy_k$. Now I'm considering the action of $S^1$ on ...
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0answers
11 views

Does the Eilenberg-Moore spectral sequence of the path space fibration of a suspension space collapse at E^2?

Let $X$ be a path-connected pointed space. Consider the path space fibration $\Omega \Sigma X \to P(\Sigma X) \to \Sigma X$ of its suspension space. Taking homology with coefficients in a field, does ...
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0answers
37 views

When is $H_i(X,Y)\cong H_i(X/Y)$?

For orientable manifolds,for what $X$ and $Y\not=\varnothing$ does this isomorphism hold true?
1
vote
1answer
78 views

how do I calculate the Euler characteristic of a Klein bottle?

I guess I have to use that it is a closed surface (manifold) and therefore its boundary is empty, so then I gotta use Gauss-Bonnet easily (but I just cannot think of how to use it.) i.e. ...
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0answers
14 views

loop homology proudct for contractible manifolds

Chas and Sullivan define the loop homology product for closed (=compact with no boundary) and oriented maniflods. Is there such loop homology product for orineted compact manifolds with boundary
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32 views

Bousfield–Kan spectral sequence for homotopy colimits

Let $\mathcal{J}$ be a small category and let $X : \mathcal{J} \to \mathbf{sSet}$ be a diagram. We define its homotopy colimit $\newcommand{\hocolim}{\mathop{\mathrm{ho}{\varinjlim}}}\hocolim X$ ...
2
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0answers
35 views

The number of bijective polynomials of particular degree in a field

I need to know please: In a finite field of q elements how many bijective polynomials exist whose degree are smaller than d?
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2answers
52 views

a question about a concept in algebraic topology

I'm reading Munkres elements of algebraic topology, and have come across a definition. Here the book shows what a link is in some example. However, I think the link of v0 should include the lines ...
0
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1answer
31 views

turning a map into a fibration

In Allen Hatcher's book Spectral Sequence page 29 Example 1.18, What means "turning the map into a fibration" and convert a map into a fibration"? Given a map $f:X\to Y$, $f$ is not necessarily a ...
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0answers
18 views

Euler Characteristic of the Barycentric Subdivision of an $n$-Simplex.

The Euler Characteristic of a Simplicial Complex is defined to be $\sum (-1)^i\alpha_i$, where $\alpha_1$ is the number of $i$-simplices in the complex. Using this formula, we can see that the Euler ...
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votes
1answer
17 views

Homotopy groups of pairs and homotopy fibration of inclusions

Let $(X,A)$ be a pair of topological spaces, where $X$ is path-connected and $A$ is a path-connected subespace of $X$ with a base point. So, we have a long exact sequence of homotopy groups $$... \to ...
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0answers
18 views

Computation of the first homology group of the sphere

I am currently trying to compute the first homology group of the sphere, using the open tetrahedron (not solid). Let $\gamma, \beta, \alpha, \rho, \delta, \nu$ represent the six edges (1-simplexes) ...
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0answers
28 views

Degree of Map between Pseudomanifold

There are two different ways to define a degree of map. Let $M$, $N$ be smooth, and $M$ is compact, $N$ is connected. If $f\in C(M,N)$, we define $\deg f$ by smooth approximation. [Milnor/Topology ...
0
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1answer
32 views

What is a thin loop?

I read one definition of a thin loop: $\gamma$ is a thin loop if there exists a homotopy of $\gamma$ to the trivial loop with the image of the homotopy lying entirely within the image of $\gamma$. ...
3
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0answers
66 views

Leray-Hirsch Using Kunneth Formula from “Differential form in Algebraic Topology” by Bott and Tu

Kunneth Formula: Let M and F are manifolds. If M has a finite good cover then $H^n(M\times F)=\bigoplus _{p+q=n} H^p(M)\bigotimes H^q (F)$ Bott and tu says One can prove Leray-Hirsch theorem by the ...
1
vote
1answer
28 views

Proof that an Infinite Simplicial Complex can only have countably many Simplices?

We define an infinite simplicial complex $K$ to be a set of simplices in some $\mathbb R^n$ sch that the following conditions hold: $1$. Given a simplex in $K$, every "face" of A (i.e., the simplex ...
0
votes
1answer
24 views

Homology of the image of a chain map vs. image of homology map

Let $C_*$ and $D_*$ be chain complexes and let $f:C_*\to D_*$ be a chain map. Since $f$ is a chain map, its image $f(C_*)$ is a subcomplex of $D_*$. My question is now the following: Assume that ...
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0answers
41 views

Group action on fibre homotopy group

Let $p : E \rightarrow B$ be a Serre fibration, with typical fibre $F \cong p^{-1}(b)=: F_b$ for each $b \in B$. I know that $\pi_1(F) \looparrowright \pi_k(F)$, but now I would like to find a way to ...
3
votes
2answers
143 views

How to prove that the quotient space of the punctured plane under dilation is homeomorphic to a torus?

Suppose we're considering the map $f:\mathbb R^2\backslash (0,0)\rightarrow \mathbb R^2\backslash (0,0)$ given by $f(x_1,x_2)=(cx_1,cx_2)$ with $c$ a positive real number. How does one show that the ...
3
votes
2answers
55 views

Simply-connected $\mathbb{Z}_p$-homology spheres?

Let $X$ be a $\mathbb{Z}$-homology $n$-sphere, i.e., a closed manifold with $\mathbb{Z}$-homology groups of the standard $n$-dimensional sphere. If $X$ is simply-connected, it is not difficult to see ...
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0answers
30 views

Invertibility of suspension in spectra

I know that spectra are supposed to be designed so that suspension is invertible up to homotopy, but I'm having trouble articulating exactly why this is the case. If $E$ is a spectrum and $\Sigma E$ ...
3
votes
1answer
40 views

Computing $H_i(\mathbb{RP}^n \times \mathbb{RP^m}; G)$

I'm trying to compute $H_i(\mathbb{RP}^n \times \mathbb{RP}^m; G)$ for $G = \mathbb{Z}, \mathbb{Z_2}$ respectively by using the cellular chain complexes. I'm not really sure how to get started, ...
2
votes
2answers
49 views

Confusion about cohomology and universal coefficients theorem.

I want to check that my understanding is correct about cohomology. Let $X$ be a topological space $G$ be an abelian group. The universal coefficients theorem, as stated in hatcher, says that the ...
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vote
2answers
28 views

How do I compute the kernel of this map?

How do I compute $\ker{(\mathbb{Z} \otimes A \longrightarrow \mathbb{Q} \otimes A)}$ where this map comes from the short exact sequence: $0 \rightarrow Tor(\mathbb{Q}/\mathbb{Z}, A) \rightarrow ...
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0answers
54 views

A Theorm on Covering spaces

I was reading a theorem,precisely Theorem 2.2.10,page 33 of this book below- ...
0
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1answer
38 views

Something about Degree of Map

I have been told that degree of map can be defined by the combinatorial method instead of the differential one. Assume $M$, $N$ is orientiable pseduomanifold, and $M$ is compact, $N$ is connected. Let ...
3
votes
0answers
16 views

pontriyagin class of quaternionic vector bundle

Let $\xi^{\mathbb{H}}$ be a quaternionic vector bundle over $X$. How to define the Pontriyagin class of $\xi^{\mathbb{H}}$ efficiently? Of course we can let $(\xi^{\mathbb{H}})_{\mathbb{R}}$ be the ...
3
votes
1answer
61 views

Role of determinant of the matrix of any Homology group.

I was thinking about the proof of the Lefschetz's Fixed point theorem and the ingeniuty of the Hopf's Trace formula, i.e. associating the trace of the matrix for deciding about the fixed points. Now ...
1
vote
1answer
33 views

Regarding the proof in “Gamelin” that any reparametrization of a path lies in the same homotopy class

I would appreciate help understanding the motivation for a line in the proof of Lemma 2.3 on page 114. The Lemma states: Let $\gamma$ be a path in $X$ from $a$ to $b$. Let $\rho$ be any map from ...
1
vote
1answer
29 views

Computation the fundamental group in two complex variables

Let $\Delta$ be the unit disk in the complex plane and $0<\epsilon <<1$. My purpose is to compute the fundamental group of the following. $X :=\{ (z,w)\in \Delta \times \Delta \; | \; zw\neq ...
2
votes
2answers
34 views

Stably equivalent bundles and stable homotopy classes of maps

We know that there is a one-to-one correspondence between isomorphism classes of principal $G$-bundles over a base space $M$ and homotopy classes of maps $M \to BG$, where $BG$ is the classifying ...
6
votes
1answer
136 views

Topological spaces with unknown fundamental group

Are there any well known topological spaces for which the fundamental group is not known yet?
0
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1answer
91 views

Creating topological spaces with portals

I'm trying to rigorously describe an object that I'm calling a "portal". The situation is easiest to describe in two dimension. I start with a line segment $pq$ in $\mathbb{R}^2$. I want to remove ...
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0answers
19 views

Make differential 1-form invariant for lift to universal cover

Suppose $ F : M \to M$ is diffeomorphism of smooth manifold $M$, and suppose $F^* \nu = \nu$ for differential 2-form $\nu$. Let $p: \tilde{M} \to M$ denote universal cover of $M$, and suppose $p^*\nu$ ...
1
vote
1answer
28 views

Examples of the Geometric Realization of a Semi-Simplicial Complex

I am reading The Geometric Realization of a Semi-Simplicial Complex by J. Milnor and here are the definitions: I find it difficult to visualize without specific examples. Can anyone help to provide ...
1
vote
1answer
43 views

homeomorphism between a boundary and a sphere

I started another question related to this. Consider $|\Delta^n|:=\{(x_0,..,x_n)\in\mathbb{R}^{n+1}:\sum_{i=0}^n x_i=1, x_i\in[0,1]\;\forall i\}$ the geometric realization of the standard n-simplex ...
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0answers
17 views

Composition of boundary homomorphism from Hatcher

We have $\partial_n ( \sigma) = \sum_{i} (-1)^{i} \sigma [ v_1, \dots, \overset { \wedge} v_i, \dots, v_n]$ and we want to show that the composition $$ \delta_n(X) \overset{\partial_n} \to ...
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0answers
59 views

Prove that the quotient map $P:G \to G/H$ is a covering space.

Let $G$ be a topological group and $H$ is a subgroup of $G$. Suppose that the subspace topology on $H$ is the discrete topology. Prove that the quotient map $P:G \to G/H$ is a covering space. My ...
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0answers
29 views

Question considering the covering map of the circle $S^1$ in Munkres, 2. edition

The map $p: R \rightarrow S^1$ given by the equation $p(x) = (\cos 2\pi x,\sin 2\pi x)$ And we are to consider the subset U of $S^1$ consisting of those points having positive first coordinate. Then ...
3
votes
2answers
58 views

Homology of $n$-sheeted covering space

Let $X$ be the Klein bottle, that is $X=\mathbb{R}^2/G$ with $$G=\langle a,b\mid a^{-1}b ab=1\rangle,$$ acting via $a: \mathbb{R}^2\to \mathbb{R}^2, (x,y)\mapsto (x+1,y)$, $b: \mathbb{R}^2\to ...
1
vote
1answer
52 views

Algebraic topology : Adam's Theorem

In class, i learn Adam's Theorem. Which states the tangent bundle on $S^n$ is trivial only $n=1,3,7$. I arise some question about $n$. Why $n=1,3,7$ are so special? The professor give simple ...
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1answer
61 views

cohomology of suspension

Let $X$ be a topological space. Let $\Sigma$ be suspension. Does $H^n(X;\mathbb{Z})\cong H^{n+1}(\Sigma X;\mathbb{Z})$ isomorphic or not? Does $H^n(X;\mathbb{Z}_2)\cong H^{n+1}(\Sigma ...
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vote
1answer
58 views

Infinite degree covering space of a bouquet of circles

I am having a hard time showing that every finite group is the automorphism group of some infinite degree covering space of a bouquet of circles (rose). Here's what I have done so far: Let $G = ...
2
votes
2answers
34 views

boundary( geometric realization of the standard n-simplex) is not equal to the geometric realization of the boundary(standard n-simplex) in general

Consider $|\Delta^n|$ the geometric realization of the standard n-simplex. I know that the $|\delta \Delta^n|=\delta|\Delta^n|$ isn't true in general, whereby $\delta \Delta^n$ is the boundary of the ...