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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3
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1answer
1k views

Understanding cohomology with compact support

I am trying to understand the definition of (singular) cohomology with compact supports. My understanding of singular cohomology goes like this. Let $X$ be a topological space. Define the singular ...
5
votes
0answers
77 views

Understanding J homomorphism

I was trying to understand J homomorphism $J:\pi_r(SO(q)\rightarrow \pi_{r+q}(S^q)$from the Wikipedia page http://en.wikipedia.org/wiki/J-homomorphism. It's clear that an element of $\pi_r(SO(q)$ ...
0
votes
1answer
49 views

Does the structure group of S^n homotopic to O(n+1)?

It is easy to show that Diff(S^1) is homotopic to O(2), but in the case of n bigger than 1 things become really complicated, I cannot see the conclusion directly.
3
votes
1answer
221 views

Intuition behind a retraction from the cylinder onto the mapping cylinder.

Please excuse me for including pictures, but I thought it was easier than trying to redraw them here. I am right now reading Strøm's book Modern Classical Homotopy Theory. I have encountered a ...
2
votes
1answer
361 views

cohomology of Eilenberg-Maclane space

In line 5, Page 394 of Allen Hatcher's book Algebraic Topology, it is claimed that $H^n(K(G,n);G)=Hom(H_n(K(G,n),\mathbb{Z});G)$ for any abelian group $G$. How to get it? I have tried but cannot ...
6
votes
3answers
991 views

A confusion about the fact that contractible spaces are simply connected

Question 1: Greenberg's Algebraic topology has a proof that contractible spaces are simply connected. In the middle of the proof, the book makes use of the following fact without justifying it ...
7
votes
1answer
2k views

Homology of wedge sum is the direct sum of homologies

I want to prove that $H_n(\bigvee_\alpha X_\alpha)\approx\bigoplus_\alpha H_n(X_\alpha)$ for good pairs (Hatcher defines a good pair as a pair $(X,A)$ such that $A\subset X$ and there is a ...
1
vote
1answer
47 views

Morphisms of complexes chain [closed]

I have a small question: Why is the following true? "If we have a continuous mapping between two topological spaces $f:X\rightarrow Y$, we can associate a morphism of chain complexes $f_*\colon ...
2
votes
3answers
295 views

A question about the proof of the fact that contractible spaces are simply connected

In greeberg's algebraic topology, the following fact is used in the proof that contractible spaces are simply connected without justification: Let $p:\mathbb{I}\rightarrow X$ be a continuous function ...
6
votes
1answer
116 views

Spaces such that $\Omega^2 X \cong X$

We know from Bott Periodicityt that there is a space X such that $\Omega^2 X \cong X$ (homotopy equivalence) , but these spaces are rather complicated and I am curious, is there any easy example of a ...
2
votes
2answers
140 views

Question on homotopy

What is the relation between the definition of homotopy of two functions " a homotopy between two continuous functions $f$ and $g$ from a topological space $X$ to a topological space $Y$ is defined ...
3
votes
1answer
74 views

Cohomology of volume forms

If g and h are Riemannian metrics on the same manifold, say both of volume 1, then it follows (I guess from Poincaré duality) that their volume forms dvol_g and dvol_h are cohomologous. Question: is ...
0
votes
1answer
54 views

Labeling edges of a cube with + and - so each face has an odd number of +s.

I am looking for a specific proof, using tools from cellular homology, of the following theorem. Let $I^n$ be the standard $n$-dimensional hypercube with its standard cellular structure. There ...
2
votes
0answers
124 views

Pushing a map off a disk

Let us assume that we have covered $\mathbb{R}^n$ with the open sets $V = 2 \cdot int(D^n)$ (the standard unit disk, and 2 means multiply the size by 2) and the family $\mathcal{U}$ of open disks W ...
0
votes
1answer
59 views

Multiplicity of a zero of an L-function and covering spaces

This question may not be suitable for MathOverflow due to its relative vagueness, hence I ask it here. I just read in Wikipedia that there was a bijective correspondence between the path connected ...
6
votes
2answers
587 views

Intuition behind definition of homotopic equivalence and distinction with homeomorphism

I am a physics student and have come across the definition of homotopic equivalence of two spaces as existence of two functions $f:X \to Y,g: Y \to X$ such that $g \circ f$ and $f \circ g$ are ...
0
votes
1answer
127 views

Why call them cycles and boundaries?

I have a small question. Why we have this designation: $n$-cycles for $Z_n$ and $n$-boundaries for $B_n$ ? Why they are called cycles and boundaries ? ...
2
votes
1answer
443 views

Meaning of Fundamental group of a graph

I am a computer science student working in graph algorithms. I am unable to understand what the fundamental group of a graph means. I have some intuition regarding the fundamental group of a ...
3
votes
0answers
107 views

Find the fundamental group and the Alexander polynomial

I would like to find the Alexander polynomial of the link $L$, described below. Let $K(q,r)$ be the $(q,r)$-torus knot embedded on a torus $V$. Inside the torus $V$, consider a smaller solid torus ...
1
vote
0answers
119 views

What's wrong with my argument to compute the vector bundles on $S^1$?

I have read the solution in Vector bundles of rank $k$ with base $S_{1}$, and I am sure that answer is correct. But I have another approach and I don't know where did I make a mistake. So as circle is ...
0
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0answers
213 views

acyclic implies identity null-homotopic?

I have proved the following for a chain complex $\mathcal{C}_{*}$ where the $\mathcal{C}_i$ are free $\mathbb{Z}$ modules, $\mathcal{C}_i = 0$ for $i>0$. The identity map on $\mathcal{C}_{*}$ is ...
2
votes
1answer
141 views

$S^n \rightarrow P^n$ the induced map on top cohomology class is zero.

How to show $f^*: H^n(RP^n;Z/2Z)\rightarrow H^n(S^n;Z/2Z) $ is zero map by definition? I know a proof by using the structure of cohomology ring of $P^n$, but the structure of the cohomology of $P^n$ ...
14
votes
1answer
393 views

What is the homotopy type of the affine space in the Zariski topology..?

I'm asking this question out of curiosity, as I was unable to come to a conclusion. Consider the affine space over an algebraically closed field, for concreteness we can work with $\mathbb{C}^{n}$. ...
4
votes
1answer
63 views

Reverse use of Seifert-van Kampen Theorem?

I am trying to use S-vK Theorem in reverse; what I know are as follows: $U$ and $V$ satisfy the requirements (open, path-connected), $U\cup V = X$, $U \cap V = N$ $\pi_1(N) = <c,d| cd=dc>$ ...
1
vote
0answers
88 views

Question on Snake lemma

we have Short exact sequence of chain complexe $0\rightarrow C\xrightarrow[]{f}D\xrightarrow[]{g}E\rightarrow 0$ i want to prove that there existe a longue exact sequence of modules $$...\rightarrow ...
2
votes
0answers
126 views

Euler characteristic of affine space

Sorry for the trivial question.. but what is the (topological) Euler characteristic of $\mathbb{A}^n$? Also, is there a genus-degree formula for affine curves similar to $g={d-1\choose 2}$ for smooth ...
1
vote
0answers
240 views

Genus of Fermat Curve

The genus of a projective Fermat curve $x^d+y^d=z^d$ in $\mathbb{P}^2$ can be computed using the formula $g={d-1\choose 2}$, where $d$ is the degree. Is the genus of the affine curve $x^d+y^d=1$ the ...
1
vote
2answers
234 views

Is there any standard N-sphere that has non-trivial first Pontryagin class?

I am wondering if there is a standard $N$-sphere that has non-trivial first Pontryagin class on its tangent bundle $TS^n$and frame bundle $FS^n$? I know that only $S^4$ has non-trivial $H^4(S^n, R)$ ...
2
votes
2answers
157 views

Extending diffeomorphism to disk

I am trying to prove the following If $f:S^1 \to S^1$ is a diffeomorphism it can be extended to a diffeomorphism $F: D^2 \to D^2$. But I can't seem to prove it. I proved it for homeomorphisms using ...
3
votes
1answer
286 views

$H_1(X,X-N,\mathbb Z_2)=\mathbb Z_2$, proof?

Given $X$ a connected manifold, and $N$ a connected codimension-1 submanifold. Can someone help me show how the result of the title holds? I tried excision. That is, I tried removing $U^c-N$ where ...
1
vote
0answers
97 views

Connected sum of two surfaces is a surface?

IS the connected sum of two surfaces a surface? Im having hard time trying to see this. Can someone kindly help me? thanks.
2
votes
1answer
344 views

Chain complex of free abelian groups splits as subcomplexes of $0\to L_{n+1}\to K_n\to 0$

The proof I am trying to get at is an exercise in Hatcher, and I can't even get it started so I'm asking just about the first part here. It's not an actual homework assignment but I've tagged it as ...
1
vote
1answer
123 views

Does every map $\mathbb{R}P^n\rightarrow\mathbb{R}P^n$ lift to a pair of maps $S^n\rightarrow S^n$?

Question: Given a continuous map $f:\mathbb{R}P^n\rightarrow\mathbb{R}P^n$, is there automatically a continuous map $g:S^n\rightarrow S^n$ such that $f,g$ commute with the covering map ...
7
votes
1answer
128 views

If compact simply connected manifold has the same rational homotopy groups as $S^n$ or $\mathbb{C}P^n$, must it have the same cohomology ring?

The question came up while trying to shorten a paper I'm writing into submission-ready length. Let $M$ be a compact simply connected manifold. By defininition, the rational homotopy groups of $M$ ...
0
votes
0answers
47 views

Intuition behind a braid operator which is also a solution for Yang-Baxter equation

I am going through this paper, 'Quantum entanglement and topological entanglement' by Louis H Kauffman and Samuel J Lomonaco Jr published in New Journal of Physics 4 (2002). It started with ...
3
votes
1answer
232 views

What is an n-oriented graph?

I was reading Allen Hatcher's Algebraic Topology, and he mentioned a 2-oriented graph while describing the covering spaces of S1∨S1. What is a 2-oriented graph? Can you give an examples of 2-oriented, ...
4
votes
1answer
192 views

Bijection between homotopy classes and basepoint-preserving homotopy classes

$[X,Y]$ is the homotopy classes of maps from $X$ to $Y$ and $[X,Y]_0$ is the based homotopy classes of based maps. If $Y$ is path-connected and $\pi_1(Y)$ is abelian, then is the inclusion $$[X,Y]_0 ...
1
vote
2answers
50 views

Seeking 'simple' space with specified homotopy

I am looking for a 'named' space $S$ such that $\pi_1(S) = \mathbb{Z}_2$ and $\pi_n(S) = \star$ (the one-point group) for all $n\geq 2$. Commentary: I know that the projective plane fits the first ...
7
votes
1answer
139 views

Why is this space aspherical?

Let $X = Y \cup Z$ be a connected, path-connected Hausdorff space. Suppose that $Y$, $Z$, and $Y\cap Z$ are all connected, path-connected, and aspherical, and that the homomorphism $\pi_1(Y\cap Z) ...
1
vote
2answers
565 views

Homeomorphic spaces have the same homology groups

How do I show that homeomorphic spaces have isomorphic homology groups? Hatcher says that it is evident from the definitions, which makes me think that I didn't understand something. How does a ...
2
votes
1answer
394 views

Difference between free homotopy and isotopy. Numer of non-isotopic curves.

I realize that whenever I think of two simple closed curves in a surface being isotopic I actually think of them as being freely homotopic (intuitively). I am really confused now. So I have the ...
1
vote
0answers
33 views

Transforming the measure in $CP^1$ mapping from Riemann sphere to $\mathbb{C}^2$-plane

I would like to know how the measure changes in $CP^1$ mapping from Riemann sphere (2-sphere) to $\mathbb{C}^2$-plane. Let a point on the 2-sphere is given by the vector ...
2
votes
1answer
50 views

What is the map $(\mathbb{C}P^\infty)^n\to G^n(\mathbb{C}^\infty)$ induced by the map $(S^1)^n\to U(n)$?

There is a map $(S^1)^n\to U(n)$, mapping each $n$-tuple to the corresponding diagonal matrix, where $S^1$ is identified with the complex numbers of unit length. There is an induced map ...
2
votes
0answers
77 views

What is the use of locally connected spaces?

One of the main properties of locally connected spaces is that their connected components are clopen and thus, they are homeomorphic to the colimit of their connected components. This is good to ...
1
vote
1answer
73 views

Is the continuous map between CW-complexes a cofibration?

If $f:A \rightarrow X$ is a continuous map between CW-complexes, then is $f$ necessarily a cofibration? I know that when $A$ is a subcomplex of $X$ and $f$ is the inclusion, the conclusion is true. ...
2
votes
1answer
91 views

Should the first be the last by composition of paths?

Given two paths $f,g:\mathbb{I}\rightarrow X$ with $f\left(1\right)=g\left(0\right)$ there is a composite $f.g$ defined by $t\mapsto f\left(2t\right)$ if $2t\leq1$ and $t\mapsto g\left(2t-1\right)$ ...
8
votes
3answers
1k views

References for Topology with applications in Engineering, Computer Science, Robotics

I am reading a book on motion planning for mobile robots, I have a really hard time with the mathematics. Some parts it is talking about the topology of the space and manifolds and compactness of the ...
4
votes
2answers
179 views

Can abstract nonsense be helpful here?

Here a question for those among you, who teach Homotopics/Algebraic Topology at university. I encountered some questions that were in my view quite easier to solve in category hTop instead of Top ...
2
votes
2answers
500 views

Equivalent definitions of “evenly covered”

I am just starting to learn Algebraic Topology and it would be very helpful to know whether the following two definitions are equivalent... Let $X$ and $Y$ be topological spaces and $p:Y \to X$ be ...
2
votes
1answer
195 views

Creating connective spectra from infinite loop spaces

I have a quick question which I think should go like this, but I am not really sure and that is why I would like someone more knowledgeable than me to weigh in and say if I am correct. Let us say ...