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

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3
votes
1answer
48 views

Weak equivalence testable on invariant open covers?

Let $f\colon X\rightarrow X'$ be a continuous map between two spaces $X,X'$, which might be arbitrary wild, especially I don't want to work in any convenient category of topological spaces. Let ...
3
votes
0answers
14 views

Is the projection from the cyclic nerve to the simplicial nerve a cyclic map?

I’m trying to understand Loday’s proof of Theorem 7.3.11 in “Cyclic Homology” (2nd ed 1998). I have a problem right at the beginning where it says: The projection map $\text{proj}:\Gamma_\bullet G ...
0
votes
1answer
17 views

Understanding a proof of lifting $F:Y\times I\rightarrow X$ to $\widetilde F:Y\times I\rightarrow \widetilde X$

The statement to prove given in Allen Hatcher's book Algebraic Topology is: Given a map $F:Y\times I\rightarrow X$ and a map $\widetilde F:Y\times \{0\}\rightarrow \widetilde X$ lifting ...
1
vote
1answer
43 views

Exercise 2 in Hatcher, section 1.2: the union of convex sets is simply connected

Here is my problem : I have convex subsets $X_1, \dots, X_n$ of $\mathbb R^m$ such that $X_i \cap X_j \cap X_k$ is never empty. I have to show that $\bigcup\limits_{i=1}^nX_i$ is simply ...
1
vote
1answer
28 views

lifting property of universal cover

I have a question. If we have a map $f:\mathbb RP^n\rightarrow \mathbb RP^n$, then can we always lift it to a map $g:S^n\rightarrow S^n$ such that the diagram commutes? $$\begin{array} $S^n & ...
0
votes
1answer
56 views

A simple question about rational Hodge conjecture

Good evening everyone : In the link here I found the following sentence : The Hodge conjecture predicts that the $\mathbb{Q}$ - linear span of the classes of algebraic subvarities in the cohomology ...
0
votes
1answer
37 views

Understanding an example from Hatcher - cellular homology

Example 2.34: An Acyclic Space says the following - Let $X$ be obtained from $S^1 \vee S^1$ by attaching two 2-cells by the words $a^5b^{-3}$ and $b^3(ab)^{-2}$. Then $d_2: \mathbb{Z}^2 \rightarrow ...
1
vote
0answers
73 views

Compute homology groups of space $\Bbb RP^2$ attached with Mobius band using Mayer Vietories

(This is exercise 2.2.28 from Hatcher) Consider the space $X$ (say) obtained from a $\Bbb RP^2$, by attaching a Mobius band $M$ via a homeomorphism from the boundary circle of the Mobius band to the ...
1
vote
1answer
59 views

$\pi_1([S^2-\{(0,0,1),(0,0,-1)\}]/x\sim -x)$ [closed]

I'm interested in calculating $\pi_1(X)=\pi_1([S^2-\{(0,0,1),(0,0,-1)\}]/x\sim -x)$ Moreover, I'm interested in $Y=[S^2\cap \{x\in\mathbb R^3:\:x_1 x_2 x_3>0\}]/x\sim -x$. Appreciate any kind ...
0
votes
0answers
31 views

How to define multiplication in covering group?

Let $G$ be a connected topological group and let $p:\tilde{G}\to G$ be a universal covering of $G$. Then $\tilde{G}$ is also a topological group and $p$ is a continuous homomorphism. My question is: ...
2
votes
1answer
27 views

Singular homology of discrete space

Let $X$ be a space with discrete topology. How to calculate singular homology of $X$, if: a) $|X|$ is finite b) $|X|$ is countable c) $|X|$ is uncountable. For $|X|=1$ it is obvious, but I have ...
1
vote
1answer
66 views

What is spectral sequence?

Can anyone explain me what is spectral sequence? What is the motivation behind this? Is it a natural tool? Why should we study spectral sequences? Pardon me for asking too many question.Actually I ...
0
votes
1answer
29 views

isomorphic fundamental groups of quotient space

I have a set $X\subset \mathbb R^n$ an equivalence relation $x∼-x$. Say $Y\cong X$ i.e they are homeomorphic. I would like to conclude $\pi_1(X/∼)=\pi_1(Y/∼)$. Is that true? It does look reasonable. ...
0
votes
1answer
15 views

Disc quotient that is homeomorphic to the pinched torus

I apologize for my previous post. There was a mistake. I want to write a quotient of the disc $D^2:={\{z\in\mathbb R^2;\parallel z\parallel \leq 1 }\}$ by an equivalence relation which is ...
0
votes
0answers
49 views

The simple meaning behind covering map & lifting

When it comes to math concepts, my sneaky feeling is that they were all started up as simple ideas, but got more and more convoluted as mathematicians tried harder and harder in making them more ...
2
votes
1answer
34 views

Homology of a space obtained from $S^n$ by attaching a cell $e^{n+1}$ by a map of degree $m$

I am trying to understand how to use cellular homology on this simple example: let $X$ be a space obtained from $S^n$ by attaching a cell $e^{n+1}$ by a map of degree $m$. I understand that the ...
1
vote
2answers
47 views

Why are the fibers of a covering map are homeomorphic?

Let $E$ and $X$ are topological spaces and $p:E \rightarrow X$ be a covering map. Whys are all the fibers homeomorphic?
2
votes
2answers
41 views

calculate $\pi_1(\mathbb D-\{(0,0)\})$

I'm interested in calculating $\pi_1(\mathbb D-\{(0,0)\})$. My guess would be that $\mathbb D-\{(0,0)\}$ is homotpic to $S^1$ and so the fundamental group would be $\mathbb Z$. Am I right? How would ...
0
votes
0answers
18 views

Criterion for a map to be homotopy equivalence in terms of its mapping cylinder

I am trying to prove that a map $f: X \to Y $ is a homotopy equivalence iff $j : X \to Z(f)$ is a deformation retract where $Z(f)$ is the mapping cylinder and $X \to Z(f) \to Y$ is the decomposition ...
1
vote
0answers
68 views

Non-vanishing winding number for the interior points of a Jordan curve.

According to the Jordan Curve Theorem if $\gamma : [0,1]\to\mathbb R^2$ is a simple closed curve, then $\mathbb R^2\!\smallsetminus\!\gamma([0,1])$ consists of exactly two connected components. One ...
1
vote
1answer
32 views

Sphere with g handles without point $\simeq \vee_{n=1}^{2g} \mathbb{S}^1$

Consider the standard representation of sphere with $g$ handles as CW-complex ($4g$-gon), obviously if we remove some point $p$, we obtain deformation retraction on border and if we factor border, we ...
8
votes
1answer
210 views

Why is the complement of a discrete subspace of $\mathbb{R}^n$ ($n \ge 3$) simply-connected?

I'm stuck with an Exercise in Hatcher's Algebraic Topology. (Exercise 1.2.6) This problem asks me to show that the complement of a discrete subspace of $\mathbb{R}^n$ is simply-connected if $n\ge 3$, ...
4
votes
0answers
76 views

Hermitian Matrices with At Most Pair-wise Eigenvalue Degeneracy

Let $n\in2\mathbb{N}$ be given. Let $H\in Mat_{n\times n}(\mathbb{C})$ be a Hermitian traceless matrix such that its eigenvalues have at most pairwise degeneracy. (That is, if the eigenvalues are ...
0
votes
2answers
28 views

Pointed maps on S^n don't associate under #.

My topology professor gave us the following definition: for two pointed maps, $f,g:S^n\to (X,x_0)$, we may regard the functions as maps on $[0,1]^n$ that are constant on the boundary. We then define ...
0
votes
0answers
30 views

Why Delta Complex structure?

I am a bit confused about delta complex structure and CW complex structrue. I was wondering is n-dimensional disks homeomorphic two n dimensional triangles? I mean if that is the case then I can make ...
2
votes
1answer
21 views

Why does this open cover of $T^n$ have intersection $T^{n-1}\sqcup T^{n-1}$?

When computing the de Rham cohomology of the $n$-torus $T^n$, usually one takes an open cover $T^n=A\cup B$, where $A=T^{n-1}\times S^1\setminus{N}$ and $B=T^{n-1}\times S^1\setminus\{S\}$, where ...
0
votes
1answer
51 views

Integral over vector field

Can someone help me with this one: Let $X$ be a vector field on $R^{3}$ such that $X(x)=x$, for each $x\in S^{2}$. Calculate $$\int\limits_{\overline{B^3(1)}} \left(\frac{\partial X_1}{\partial x_1} ...
0
votes
0answers
38 views

CW-complex definition

I've the following definition a (finite) CW-complex of dimension $N$ is a topological space $X$ defined in the following way: $X^0$ is a discrete set of points $\forall 0<n\le N$, $X^n$ is ...
7
votes
3answers
128 views

Prove that $\mathbb R ^n $ without a finite number of points is simply connected for $n\geq 3$

I want to prove that $\mathbb R ^n $ without a finite number of points is simply connected for $n\geq 3$. Let $X$ be that finite set of points. My idea is to prove this by induction on cardinality of ...
0
votes
0answers
67 views

characteristic classes of $SO(3)$-bundles over $\mathbb{CP}^2$

Let us consider the complex line bundle $\xi$ over $\mathbb{CP}^2$ which is completely defined by its restriction on a complex projective line; this restriction is denoted by $\xi^{\prime}$ and the ...
0
votes
1answer
66 views

proving the injectivity half of de Rham's Theorem when $p>1$ (if a $p$-form $\omega$ vanishes on all $p$-cycles, then $\omega$ is exact)

Let $M$ be a smooth manifold of dimension $n$, and let $\omega$ be a differential $p$-form on $M$. Then we have the following theorem: $\omega$ is exact if and only if $\oint_c\omega=0$ for all ...
0
votes
0answers
69 views

Prerequisite to start learning the Fulton's book about : Intersection theory. [duplicate]

Good evening everyone , Could you tell me please, what to have as a prerequisite to learn the following course here [link removed by a moderator, because at least two users expressed their concern ...
1
vote
3answers
74 views

A cylinder is not homeomorphic to a disk

A teacher told us that a cylinder can't be homeomorphic to a disk, but he was unable to give a 'simple' proof; the only proof he knew uses the fundamental group which I've never seen in my life. Any ...
2
votes
2answers
75 views

Proving that a fundamental group is uncountable

Given the space $\mathbb{R}^2 - \bigl(\{0\}\times\mathbb{Q}\bigr)$, I need to show that the fundamental group of this space is uncountable. I thought of taking two points $A=(x_0,y_0)$ in the area ...
2
votes
1answer
55 views

Inverses of homotopic paths are homotopic too

PROVE: If $g_1 \simeq g_2$, then $\bar{g_1} \simeq \bar{g_2}$. I found the solution online here, which I copy it down here for convenience: (1) Let $\bar{g_1} : I \to X$ be defined by ...
1
vote
0answers
66 views

Covering between universal covers

While trying to solve a problem, an intuitive idea has brought me to the following statement. Is it true? If yes, how can we prove it? If $X$ is a covering space of $Y$, then the universal cover of ...
1
vote
0answers
42 views

Homology of Tori Glued by a Matrix

I'm having trouble calculating the homology groups for a certain space that's hard for me to envision. Let $X$ be obtained by taking two solid tori and gluing them together along their boundaries ...
0
votes
0answers
23 views

Uniqueness for a covering map lift: is locally connected necessary?

So I just got through proving the following theorem: If $p:C\to X$ is a covering map and $Y$ is a [xxx] space, then given $y_0\in Y$, $c_0\in C$, $f:Y\to X$ such that $f(y_0)=p(c_0)$ there exists ...
1
vote
0answers
35 views

homomorphism inducing Galois cover

We are given a homomorphism $\rho: \pi_1(\Sigma_g - \{p_1,...,p_k\}) \rightarrow G$ , where $\Sigma_g$ is genus g Riemann surface and $p_i$'s are points on it, G is a finite group. Then it is claimed ...
1
vote
1answer
23 views

Simplicial homology of sphere with bars.

Let $X$ be a space obtained by inserting $n$ vertical bars into $S^2$. I want to compute simpicial homology of this space. I am going to start with just one bar, the triangulation I have is the ...
5
votes
1answer
78 views

Intuition for an abelian fundamental group

Any topological group has an abelian fundamental group by the Eckmann-Hilton argument. Is there some intuition behind the fundamental group being abelian that would enable one to predict this ...
1
vote
1answer
43 views

Suppose $X_1, X_2$ are to path connected spaces. I want to show that $X_1 \times X_2$ is path connected.

Suppose $X_1, X_2$ are to path connected spaces. I want to show that $X_1 \times X_2$ is path connected. Let $(x_1,y_1)$ and $(x_2, y_2)$ be given. By assumption i know there exist: $\gamma_1: ...
0
votes
0answers
24 views

cohomology of local system

Let $X_r$ be a finite simplical complex. Let $V_r$ be a sheaf which is a local system on $X_r$. Is it true that: $H^n(X_r,V_r$) i.e the cohomology of the sheaf $V_r$ coincide with the cohomology of ...
1
vote
2answers
56 views

Hatcher 2.2.26 Show that if $A$ is contractile in $X$ then $H_n(X,A) =H_n(X) \oplus H_{n-1} (A)$

Show that if $A$ is contractible in $X$ then $H_n(X,A) \approx \tilde H_n(X) \oplus \tilde H_{n-1}(A)$ I know that $\tilde H_n(X \cup CA) \approx H_n(X \cup CA, CA) \approx H_n(X,A)$. And $(X ...
1
vote
2answers
96 views

Is the interval [0, 1] in homotopy subset of real number?

I am asking this question within the framework of my earlier posting here, for which I am grateful to @aes and @kobe for their generous help, thank you! Here are the question and its solution again ...
0
votes
2answers
78 views

Fundamental group via Van Kampen

I want to compute the fundamental group of the set C defined below : $A_{1}:=[0,1]²,A_2:=[-1,0]\times[0,1],C=\partial A_1 \cup \partial A_2$. I have to use the Van Kampen theorem and so I know that ...
0
votes
1answer
57 views

Bundles with nonzero Stiefel class over $\mathbb{CP}^2$

Could you please show me an example of a principal $SO(3)$-bundle over $\mathbb{CP}^2$ such that the restriction of this bundle on a projective line is non-trivial. Edit: One can try to construct the ...
4
votes
1answer
33 views

Description of real projective space $P^3$

I know that the real projective plane $P^2$ can be thought of as a union of a mobius band and a disk, where the union occurs among the common boundary of the two (circle). My question is about $P^3$. ...
2
votes
0answers
29 views

Kan's loop group construction

I'm looking for a good place to read about the loop group construction $G : \mathbf{sSet_0} \to \mathbf{sGrp}$ taking a reduced simplicial set $X$ and producing a simplicial group $GX$. I would also ...
1
vote
0answers
45 views

Calculation of Poincare dual to $\mathbb{CP}^n$ in $\mathbb{CP}^{n+1}$

I would like to go about finding an explicit representative of the Poincare dual to $\mathbb{CP}^n$ in $\mathbb{CP^{n+1}}$, I am following Bott & Tu and would like an explicit form to be wedged ...