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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1answer
676 views

The double cover of Klein bottle

I try to find out all the double covers of Klein bottle. Since the Euler characteristic is multiplicative with respect to covering space, there are only two candidates, that is, torus and Klein bottle ...
5
votes
3answers
5k views

Homology of the Klein Bottle

I know that in general, $H_{n}(X)$ counts the number of $n$-cycles that are not $n$-boundaries of a simplicial complex $X$. So for the sphere, $H_{0}(X) \cong \mathbb{Z}$ since it is connected. Also ...
2
votes
1answer
17 views

embeddability of connected sum of surfaces

Let $X$ be a surface which can not be embedded in n-space. Let $X \# X $ denotes the connected sum of two copies of $X$. Then is it true that the connected sum $ X \# X $ is also not embeddable in ...
2
votes
1answer
34 views

Examples with zero first Stiefel-Whitney class and nonzero second Stiefel-Whitney class

What's the simplest/most concrete vector bundle you can think of that has zero first Stiefel-Whitney class but non-zero second? That would be the simplest space that doesn't have spinors. (See Spin ...
2
votes
1answer
45 views

simply connected covering of a path connected space

Let $p:\overline{X}\rightarrow X$ be a simply connected covering of a path connected space $X$ and $A\subset X$ be a path connected set. Show that the inclusion induced homomorphism $i_{\sharp} : ...
1
vote
0answers
11 views

Morphism of modules of sections of pullback bundles

Suppose that we have a morphism $\theta: \Gamma(B,E_1) \to \Gamma(B,E_2)$ where $E_i$ are two vector bundles over $B$ and let $f:A \to B$ be a continuos map. Then we can define a pullback bundles ...
0
votes
1answer
9 views

Section of pullback bundle

Suppose that $E \to B$ is a vector bundle and $f:A \to B$ is continuos. If $s$ is a section of $E$ how to define a section of pullback bundle? On wikipedia they say that it induces the section of the ...
1
vote
2answers
29 views

Show that the Möbius band has its central circle $C$ as a deformation retract

I have started this problem by using the planar representation of the Möbius band and noted that a line down the middle is probably what is meant by the central circle, since travelling from top to ...
2
votes
0answers
23 views

A problem on covering space from Hatcher book…

I was trying a problem from Hatcher's books in section 1.3 problem number 12. I know corresponding to every subgroup of this group there exist a covering space.One way to find covering space ...
0
votes
0answers
23 views

“Cohomology classes correspond to homotopy classes of maps to Eilenberg Maclane spaces” and cup product?

I read this in Hatcher. I am especially interested in knowing if the cup product can be understood from this perspective? I would appreciate a reference.
2
votes
0answers
33 views

Let $A$ be the annulus in $\mathbb{C}$, what is the space $A/{\sim}$ generated by identifying all points on the “inner circle” with each other?

The annulus is $A=\{z\in\mathbb{C}:1\leq |z| \leq 2\}$ and the "inner circle" here is the set of points $\{z\in\mathbb{C}:|z|=1\}$. If we identify all points of the inner circle together, then, ...
1
vote
1answer
24 views

About covering spaces

Suppose X is a topological space whose fundamental group is Z x Z x Z2 x Z3. Is it possible for the wedge sum of two circles to be a covering space for X? Can anyone help me with this ?
2
votes
0answers
41 views

Embed $S^{p} \times S^q$ in $S^d$?

Can we embed $S^{p} \times S^q$ in $S^d$ with all the nice properties, what are the allowed values of $p$ and $q$ for $d=2,3,4$ where $p+q \leq d$? =For $d=2$= I suppose that we cannot embed $S^{1} ...
2
votes
3answers
55 views

How many elements does the free product $\mathbb{Z}/2\mathbb{Z}\ast\mathbb{Z}/2\mathbb{Z}$ have?

Taken from Hatcher, I think the free product $G =\mathbb{Z}/2\mathbb{Z}\ast\mathbb{Z}/2\mathbb{Z}$ should have infinite elements taking the form of words containing alternate elements, i.e. $a, b, ab, ...
5
votes
2answers
140 views

Homology and (co)Limits

I've looked around on MSE and online only to find scattered results, which confuse me. I want to understand how homology behaves with (co)limits. I want to know in particular about singular homology, ...
-11
votes
0answers
51 views

i want to know about cube root of unity [on hold]

Blockquotecomlex number cube root of unity origin of comlex numbers origin of cube root ofunity
0
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0answers
16 views

Questions about strata of a variety: the nilpotent cone of $\mathfrak{sl}_2$.

I am reading the lecture notes. In the end of page 3, let $X = \{(a, b; c, -a): a, b, c \in \mathbb{C}^3, a^2 + bc = 0\}$. It is said that there are two strata: the regular orbit $U$ and $0$. What is ...
1
vote
1answer
53 views

On the associative property of a binary operation of the fundamental group.

I was reading about the proof of associativity property of the operation on the fundamental group here. The book gives the following diagram then it says the reader should supply the elementary ...
0
votes
3answers
33 views

Relative homology $H_n(S^2,S^0)$, or other examples

I've been reading Hatcher and think I understand the idea of relative homology, but he only provides two (fairly trivial) examples, homology relative to a point computing $H(S^n)$ using $D^n$s. My ...
3
votes
3answers
54 views

The significance of filtered colimits in homotopy theory

I have been allowed to attend some preparatory lectures for a seminar on the Goodwillie Calculus of Functors. I found in my notes from one of the lectures two statements which I would like to ask ...
5
votes
1answer
58 views

fiber bundle in topological category and smooth category.

Let $M$ be a smooth manifold and $G$ be a Lie group. Denote by $Bun(M,G)$ the set of all equivalent smooth Principal bundle on $M$ with structural group $G$ in smooth category. And denote by ...
0
votes
0answers
129 views

Making my Topology joke mathematically rigorous. [closed]

I think I have fairly good material, unfortunately I'm not profficient enough in topology to be sure it is mathematicaly correct. Here is my joke in its current form: $\color{purple}{\text{How does ...
4
votes
1answer
60 views

Do join and suspension commute?

Do join and suspension of topological spaces always commute, i.e. is it true that $\sum(A\star B)=A\star(\sum B)$? I suppose that it is not true in general (but, for example, everything works in the ...
3
votes
1answer
28 views

A question about the degree of a map

Consider $f,g: S^{n}\rightarrow S^{n}$ to be continuous maps. Seeing $S^{n}\subset\mathbb{R}^{n+1}$, let's say that $f$ and $g$ are orthogonal at $x\in S^{n}$ whenever $\langle f(x),g(x)\rangle = 0$ ...
1
vote
1answer
21 views

Is a covering space of a topological group a fiber bundle with the structure group the fundamental group of the topological space?

Question. Is a covering space of a topological group a fiber bundle with the structure group the fundamental group of the topological space? Let $p:E\rightarrow X$ be a covering space of X. I ...
1
vote
0answers
13 views

Conjugate group homomorphisms induce homotopic maps on classifying spaces

Let $G$ be a group and let $\phi: G \to G$ be the inner automorphism given by conjugation by an element $g' \in G$, i.e., $\phi(g) = g'^{-1} g g'$. I want to show that the induced map on classifying ...
3
votes
0answers
55 views
+50

Textbook on infinite loop spaces

I'm looking for a good update reference covering the material in first three chapters of "Adams, Infinite loop spaces" (specially construction of delooping functors and group completion) with exact ...
1
vote
1answer
36 views

Fiber bundle beginner question.

I'm reading some notes on fiber bundles. Let $f:X \rightarrow Y$ be a continuous map of topological spaces. The author states: We say $f$ makes $Y$ a fiber space over $X$ if $f$ is locally trivial ...
4
votes
1answer
116 views

A question on Hawaiian earring

Consider the Hawaiian earring. Suppose $f_n$'s are the loops representing the circles of radius $1/n$ centered at $(1/n, 0)$ for $n=1,2\cdots$. Suppose the following holds in the fundamental group ...
9
votes
1answer
137 views

Prove $\mathbb{R}^3$ is not the product of two identical topological spaces

I can only prove this for $\mathbb{R}$: If $\mathbb{R}\cong T\times T$, then $T$ embeds in $\mathbb{R}$ as a closed subspace (e.g. $T\times pt$). Since $\mathbb{R}$ is connected, so is $T$. So $T$ ...
7
votes
2answers
106 views

Fundamental group of a quotient on a solid torus.

It is easy to compute the fundamental group of a solid torus. You easily get $\mathbb{Z}$ just because the torus is the cartesian product of a circumference and a closed disk. The next step is ...
-1
votes
0answers
62 views

How to show $S^n$ is not contractible without using Homology..

I know the prove $S^n$ is not contractible using homology.But I don't know how to prove it from definition of contractibility.Can anyone help me in this direction? Thanks.
2
votes
1answer
69 views

Shrinking wedge of circles

I'm spending too much time thinking about this problem : I need to show that the shrinking wedge of circles which is path connected, locally path connected ,doesn't have a simply connected covering ...
3
votes
3answers
59 views

The biggest degree of a map between fixed surfaces

Let $S_1$ and $S_2$ be compact surfaces (real manifolds of dimension 2). None of them is a sphere. Question: What is the biggest degree of a smooth map from $S_1$ to $S_2$? Comment 1. I have a ...
-1
votes
1answer
44 views

classification theorem in a subset of R^2

I need some very simple results of algebraic topology but I am not sure where I can find them without having to swallow the whole theory. What I want: -An open bounded subset $A$ of $R^2$ is ...
0
votes
2answers
102 views

Finite fundamental group and covering spaces

Show that if a path-connected, locally path-connected space X has a finite fundamental group , then every map $X$ to $S^1 \times S^1$ is nullhomotopic (i.e. homotopic to a constant map) . Is the ...
13
votes
1answer
300 views

Maps from $D^n$ to $D^n$ with a single inverse set are open.

Let $D^n$ denote the closed unit ball in $\Bbb R^n$. In multiple sources proving Brown's generalized Schoenflies theorem (including a version in the original paper), the following consequence of ...
2
votes
1answer
47 views

Calculating $H_1(\mathbb{R})$

Given the space $X=\mathbb{R}$, how can we calculate its first homology group $H_1(\mathbb{R})$? Intuitively, the object of first homology describes 1-dimensioal holes in the set which here doesn't ...
-1
votes
0answers
22 views

Finding whole number coordinates on continuous curves [on hold]

Imagine two surfaces in 3D space defined by known equations intersect and form a line in 3D. How could you find out if that curve formed by the intersection goes through any points where all 3 ...
2
votes
3answers
40 views

Difference between Wedge of countable infinite circle and Hawaiian ear ring?

Hawaiian ear ring is the union of countable circles at points (0,1/n) with radius 1/n.It seems to me that wedge sum of countable infinite circle is same as Hawaiian ring.But I found that this not ...
1
vote
1answer
61 views

Is there a more general obstruction to the existence of moduli spaces than the existence of automorphisms?

We are taught that, in general: A type of objects that has nontrivial automorphisms cannot have a fine moduli space. The proof generally goes along the lines of: Take an object $X$ with a ...
2
votes
3answers
37 views

locally path connectedness

While studying covering spaces , hatcher mentioned the "shrinking wedge of circles" this space is locally path connected as I was told , but I wasn't able to prove it nor to see it, it looks like comb ...
2
votes
1answer
50 views

Universal covering of the complement of a circle in $\mathbb{R}^3$

What is the universal covering of $X=\mathbb{R}^3\setminus(S^1\times\{0\})$? I've been trying to build a covering map from $\mathbb{R}^3$ onto $X$ via composition of $p:\mathbb{R}^3\to Y$ and $q:Y\to ...
2
votes
1answer
50 views

Proving that $H_0(X)=\tilde{H_0}(X)\oplus\mathbb{Z}$

In the article about the Reduced Homology it's stated that $$H_0(X)=\tilde{H_0}(X)\oplus\mathbb{Z}$$ but I don't know how to prove that. I know $$H_0(X)=\bigoplus_{\alpha\in ...
-1
votes
1answer
26 views

Covering Space of Orthogonal Group [on hold]

What is the covering space of Special Orthogonal Group SO(3)?
4
votes
0answers
28 views

How do we see the rank of the braid group?

The only presentation of the braid group that most people ever see is the standard Artin presentation $$B_n = \langle \sigma_1, \cdots, \sigma_{n-1} | \ \ \sigma_i\sigma_j = \sigma_j\sigma_i\ \ (|i - ...
1
vote
0answers
55 views

Finest good cover of a topological space

Let $X$ be a topological space. Does there exists a good open cover $\left\{ U_{a}\right\}_{a\in I}$ finer than any other open cover of $X$? A good cover $\{U_\alpha\}_\alpha$ of $X$ is a ...
0
votes
2answers
34 views

Show the two fundamental groups representation of the Klein bottle are the same

Show the two different representation of the fundamental groups of the Klein bottle $G=<a,b:aba^{-1} b>$ and $H=<c,d:c^2 d^{-2}>$. As we calculate in different and gett one result as ...
-1
votes
1answer
49 views

Induced homomorphism - Homology [on hold]

I have some difficulty understanding what is a homomorphism induced by a certain function f. For example, let f: X->Y a function such that f(x)=y. Now, suppose H⊂X et H'⊂Y, hence f induces a ...
2
votes
0answers
31 views

How to compute perverse sheaves?

In the video, from 49:00 to the end of the video, there is an example of computing $IC(S, L)$ and equivariant local systems. I don't understand some parts of the computations. Let $X$ be the variety ...