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

learn more… | top users | synonyms (1)

2
votes
0answers
12 views

Applications of Topological Complexity of configuration space

I'm starting to work on Topological Complexity of configuration spaces . Articles say that it has applications in robotic and control theory . My questions are : 1) How Topological complexity can help ...
4
votes
0answers
44 views

Is there a general way to tell whether two topological spaces are homeomorphic?

We know that if two topological spaces $X$ and $Y$ are homeomorphic, then they have the same fundamental groups, and the same homology. In other words, we have functors $$\pi_1 : \mathsf{Top} \to \...
0
votes
2answers
43 views

Strong Topology is the strongest topology?

In his article Construction of universal bundles. II (1956), John Milnor defines the strong topology in a join of spaces, but his definition is By a strong topology in $A_1\circ A_2\circ \dots \...
3
votes
1answer
30 views

Construct smooth mapping $f: B^{n + 1} \to S^n$ with two singularities at which $f$ has degree $+/- 1$.

I'm currently working through a paper by Pjotr Hajlasz who wants to show that For smooth manifolds $M,N$, if $\pi_{[p]}(N) \neq 0$ and $1 \leq p < n = \dim M$, then the smooth mappings $C^\...
1
vote
1answer
33 views

Topological Join of Unit Balls

I have seen that apparently one has for spheres that $S^n*S^m=S^{n+m+1}$. Is there a similar result for unit balls? Thank you.
2
votes
1answer
72 views

Show that the homology induced homomorphism $f_*:H_3(RP^3)\rightarrow H_3(S^2\times S^1)$ is a zero map.

Let $f:\mathbb{RP}^3\rightarrow S^2\times S^1$ be a continuous map. Prove that induced map $f_*:H_3(\mathbb{RP}^3)\rightarrow H_3(S^2\times S^1)$ is a zero map. I found that the third homology of ...
0
votes
1answer
30 views

Interpretation of points in covering spaces as homotopy classes of paths [on hold]

If $p:\widetilde{X} \to X$ is a covering map, $y \in \widetilde{X}$ determines a homotopy class of paths in $X$ joining the base point $x_0$ to the point $p(y)$. But a homotopy class of paths in $X$ ...
6
votes
1answer
62 views

For $n\geq 2$, any continuous map $f:\mathbb{C}P^n\rightarrow S^2$ induces the zero map on $H_2(*)$

I am working through an old qualifying exam from another university. My course did not cover as much material as what is on this test (e.g. we did not cover cohomology). So I am just working through ...
7
votes
1answer
87 views

Why the attachment to simplices in (co)homology?

I've been thinking a bit about why we define the singular homology and cohomology groups with simplices rather than, say, cubes, and it seems to me that the elementary aspects of the theory would all ...
3
votes
1answer
59 views

If two maps are homotopic, are the images homotopy equivalent?

My question is; If two continuous maps $f,g:X\rightarrow Y$ are homotopic, are the images $Im(f),Im(g)$ homotopy equivalent? Clearly, the converse is false. If it is false, is there any condition ...
0
votes
1answer
20 views

Question About Covering Space Classification Theorem

I'm a bit confused by Hatchers choice of words here. He says "The main classification theorem for covering spaces says that by associating the subgroup $p_{*}(\pi_{1}(\tilde{X},\tilde{x_{0}}))$ we ...
2
votes
2answers
65 views

Fundamental group contains $\mathbb{R}$ or $\mathbb{Q}$

Is there any topological space whose fundamental group contains $\mathbb{Q}$ or $\mathbb{R}$? In case of (singular) homology or cohomology, we can change its coefficients to any abelian groups (with ...
0
votes
0answers
25 views

A doubt in Whitehead's proof about cohomology with local coefficients [on hold]

In the proof of Theorem 4.9 says that $p^*:H^n(X_n,X_{n-1};G|X_n) \to Hom(H_n(\widetilde{X}_n, \widetilde{X}_{n-1}),G_0)$ has image $Hom^{\pi}(H_n(\widetilde{X}_n, \widetilde{X}_{n-1}),G_0)$. ...
2
votes
1answer
28 views

Map inducing zero on first cohomology is nullhomotopic (plus assumptions on fundamental group and universal cover)

Currently I am studying for a topology exam next week and came across an exercise where I could need some hints (cf. here): Let $X$ be a path-connected space with $\pi := \pi_1(X,*)$ abelian and ...
2
votes
1answer
62 views

Counter-example for $\tilde{H} (X/A) \cong H (X, A)$?

Yo! I was not able to find a counter-example to $$\tilde{H} (X/A) \cong H (X, A)$$. It's a well known fact that for cofibrations $A \hookrightarrow X$ (or more generally whenever $A$ is a deformation ...
1
vote
0answers
18 views

Explicit isomorphism $H (E, E^0) \cong H_{cv} (E)$

Yo! I've been trying to understand better why $$H (D, D-\{x\}) \cong H_c (D)$$ for a disk $D$ and, more generally, why $$H (E, E^0) \cong H_{cv} (E)$$. In general, the first isomorphism can be seen as ...
0
votes
0answers
15 views

Product and Join of $G$-CW-Complexes

Given a topological group $G$ and two $G$-CW-Complexes $X$ and $Y$ I want to understand the natural CW-structure on $X\times Y$ and $X*Y$. I understand that the concepts are very similar, so I want to ...
3
votes
0answers
48 views

Why all differentials are $0$ for Serre Spectral Sequence of trivial fibration?

Consider the fibration $F \hookrightarrow F \times B \to B$. I understand that if I take kunneth's theorem for granted that the group extensions $F_{n-i,i} \to F_{n-i+1,i-1} \to F_{n-i+1,i-1}/F_{n-i,...
2
votes
1answer
47 views

Fixed-point free map of the 2-sphere which has order 4

The antipodal involution of $\mathbb{S}^2$ clearly has no fixed points. However I cannot think up an example of homeomorphism of order 4 which has no fixed points. Could you help me?
1
vote
1answer
22 views

Steenrod Operations an algebraic Approach

Assume that $q=p^{r}$, where $p$ is a prime either 2 or odd and $\mathbb{F}_{q}$ is a Galois field and $V$ a finite dimensional $\mathbb{F}_{q}$-vector space. Then due to Larry Smith in this http://...
1
vote
2answers
57 views

How to show that $\pi_1(\mathbb{RP}^2) = \mathbb{Z}_2$?

How to show that $\pi_1(\mathbb{RP}^2) = \mathbb{Z}_2$? In general is well known that $\pi_1(\mathbb{RP}^n) = \mathbb{Z}_2, ~ n \ge 2.$ But how to show this assertion? I have a few knowledge about ...
1
vote
2answers
23 views

A relation between interior and closed sets

A topological space $X$ is said to be completely regular provided that it is a Hausdorff space such that, whenever $F$ is a closed set and $x$ is a point in its complement, there exists a function $f\...
1
vote
1answer
36 views

If 2 loops with equal base points are homotopic, must they be homotopic relative to the base point?

Let $X $ be a topological space and $\mathbb {S}^1$ be the set of complex numbers with magnitude 1 equipped with the inherited topology from $\mathbb {C} $. If we have 2 loops $f,g:\mathbb {S}^1\...
1
vote
1answer
39 views

Can we lift paths of a Lie group quotient $G\to G/H$?

Question: Let $G$ be a Lie group and $H\subseteq G$ a closed normal subgroup. Let $$\pi:G\to G/H$$ be the quotient map. If $\gamma:[0,1]\to G/H$ is a smooth path, can we find a smooth path $\tilde{...
1
vote
0answers
21 views

basepoint problem: Is there an action of $\pi_1(B)$ on $\pi_1(F)$ for $F$ path connected

I am doing this to try to figure out The action of $\pi_1(BK) \curvearrowright H_*(BG)$ for the fibration $BG \hookrightarrow BH \to BK$ . Let the fibration $F \hookrightarrow E \xrightarrow{p} B$ be ...
2
votes
0answers
25 views

If $p: E \rightarrow X$ is a covering map with $E$ connected and $|p^{-1}(x_{0})|=k$ for some $x_{o}$ then $|p^{-1}(x)|=k$ for all $x \in E$.

Prove that if $p:E \rightarrow X$ is a covering map with $E$ connected and $p^{-1}(x_{0})$ has $k$ elements for some $x_{0} \in X$, then $p^{-1}(x)$ has $k$ elements for every $x \in X$. Is my proof ...
1
vote
1answer
32 views

verifying homeomorphism of orbit space and suggestions for further study

Define an action of $\mathbb{Z}_2$ on $S^1$ by $(0,z)\mapsto z$ and $(1,z)\mapsto \bar{z}$. An orbit of $z$ is then the set $\{z,\bar{z}\}$. I claim the orbit space $S^1/\mathbb{Z}_2$ is homemorphic ...
2
votes
2answers
45 views

Principal bundle as homotopy fiber universally self-trivializes

In this MO answer, I was told the definition of principal bundle as a homotopy fiber of its classifying map precisely says that it's the universal bundle which trivializes itself. However, I'm having ...
8
votes
2answers
85 views

Homology and cohomology of 7-manifold

I have the following problem: Let $M$ be a connected closed $7$-manifold such that $H_1(M,\mathbb{Z}) = 0$, $H_2(M,\mathbb{Z}) = \mathbb{Z}$, $H_3(M,\mathbb{Z}) = \mathbb{Z}/2$. Compute $H_i(M,\...
0
votes
0answers
24 views

Two sheeted covering projection

In Hatcher on page 144, example 2.42, I see $RP^n$ described as a CW structure with one cell $e^k$ in each dimension $k\leq n$, and the attaching map for $e^k$ is the 2-sheeted covering ...
2
votes
0answers
49 views

Cohomology Group basis

I'm reading a text on Complex Torus and Abelian Variety and at a time is written as follows: The cohomology group $H^{1}(T,\mathcal O_{T})$ has a basis $w_{j}=d\overline{z}_{j}, j=1,2,...,g,$ as ...
2
votes
1answer
43 views

Restriction of a $C^{\infty}$ vector bundle over a regular submanifold.

This question is about the content of page 134 of Tu's An Introduction to Manifolds. A $C^{\infty}$ vector bundle or rank r is a triple $(E,M,\pi)$ consisting of manifolds $E$ and $M$ and a ...
-1
votes
0answers
28 views

Universal Abelian Covering Space of genus two surface [closed]

Let M be a surface M, i am concerned with Abelian covers. These are the covering spaces for which the deck group is Abelian. The largest such cover corresponds to the commutator subgroup of the ...
1
vote
2answers
29 views

Let $X$ be the union of a torus with an interval that meets the torus as shown. Use Van Kampen to find a presentation for the group.

I need to come up with some kind of cell structure here right? How can I do this?
1
vote
1answer
42 views

Classifying space of $GL_{n}(\mathbb{F})$?

I was looking for the classifying space of the general linear group $GL_{n}(\mathbb{F})$ over a field (of characteristic either zero or positive, finite or infinite), but unfortunately I didn't manage ...
0
votes
0answers
24 views

The action of $\pi_1(BK) \curvearrowright H_*(BG)$ for the fibration $BG \hookrightarrow BH \to BK$

**Note:I was extremely confused when I wrote this post. Please see the linked one. I left this one as it is, because what I understand now is so radically different then what I wrote below ** Let $...
0
votes
0answers
31 views

Does limit functor preserve isomorphism in inverse systems?

Let $I$ and $I'$ be an inverse systems for which limit exists (For example R-modules) spanned by some indexing categories with order-relation $\lambda$ and $\lambda'$ respectively. Let $\phi:\lambda' \...
3
votes
2answers
62 views

Topology of complex projective plane

It is well known there are two ways to construct topology of $\mathbb{C}P^n$: quotient space of $S^{2n+1}$ by identifying $x$ with $\lambda x$, where $\lambda$ is complex nonzero constant. According ...
3
votes
1answer
73 views

Action of $\mathbb{Z}/3\mathbb{Z}$ on $P^{1}$

I am reading from the book Topics in Galois theory by Serre. I have the following question , take $G=\mathbb{Z}/3\mathbb{Z}$. The group $G$ acts on $P^1$ by $$\sigma x\;=\;1/(1-x)$$ where $\sigma$ ...
6
votes
2answers
223 views

Intuition behind CW complexes

These things are the bane of my existence in mathematics. I feel that I can't find any clear examples of these things anywhere. This is a vague question, but how exactly do we intuitively visualize ...
1
vote
0answers
45 views

Fundamental group of hole-punched torus with boundary identification

I'm trying to find the fundamental group of $Y$ obtained from the torus by removing a small disk and identifying the boundary with the torus meridian. Here's my idea. The torus has the polygon ...
1
vote
1answer
31 views

Fundamental group of $S^{1}$ unioned with its two diameters

Is my solution correct? Call $X$ the riflescope space (I made this name up). I let $p$ be the point of intersection of the two diameters, and $q$ be the right point of intersection of the horizontal ...
0
votes
0answers
19 views

Obstruction theory vs. homotopy lifting property of Serre fibration

The obstruction to obtaining a lifting to the total space $E$ of a Serre fibration $E \to B$ of a map $X \to B$ can be derived by assuming a CW complex structure for $X$ and examining the obstruction ...
2
votes
0answers
32 views

Group bundles for topological spaces without universal cover

I‘m currently writing my Bachelor Thesis on (Co-)Homology with local coefficients. In this context I have two related questions: There are two approaches in defining Homology with local coefficients ...
1
vote
1answer
39 views

Bott and Tu compact cohomology of the circle “differential forms in Algebraic Topology”

On page 27 of that book, it is claimed that the inclusion map $\delta$ which maps a form from the non-empty intersection of two open covers of the circle to the disjoint union of those covers has a ...
0
votes
0answers
53 views

Use van Kampens theorem to compute the fundamental group of a torus with a ball attached via a map

Use van Kampens theorem to compute the fundamental group of the following space: $A$ is a torus with an open disk $D$ removed. Let $f:\partial B \rightarrow \partial A$ be a map from the boundary of a ...
2
votes
1answer
53 views

Local Properties of Immersions and Submersions

This is from Bredon's Topology and Geometry, pp 83~83. He's definitions of immersion and submersion are the following: Now, in 7.3 Theorem, he explains that (using the implicit function theorem) if ...
0
votes
0answers
34 views

branched cover of projective line over rationals

I was reading something when I came across the following phrase "branched cover of projective line over rational " . To understand what does author mean , I started reading , Now I know about ...
0
votes
1answer
33 views

Fundamental group of quasi circle(page 79 ,Hatcher) is trivial

I was trying to show that fundamental group of quasi circle(page 79 ,Hatcher) is trivial.I can understand that every loop is precisely zero loop because for any loop if it start at some point it has ...
4
votes
2answers
168 views

Broken line is NOT diffeomorphic to the real line

This is from Bredon's Topology and Geometry, page 71. This comes right after the very definition of differentiable manifold, so I think no use of tangent space or 'differential' is permitted. (Bredon ...