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

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Question About Transgression

I have been working on this question here. Here is the setup: First, all cohomology groups are assume to be with $\mathbb{Q}$ coefficients. We assume that $H^*(K(\mathbb{Q},n))=\mathbb{Q}[x]$, with ...
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1answer
85 views

More interesting examples of spaces that are retractions?

I learned about retraction: A continuous map $r: X \to A$ where $A$ is a subspace of $X$ is called a retraction if $r|_A = id_A$. I made some examples. For example: If $D$ is the closed unit disk and ...
6
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2answers
603 views

Function doesn't have a lift in a space related to Topologist's sine curve

I'm trying to solve exercise 1.3.7 in Hatcher's Algebraic Topology: Let $Y$ be the quasi-circle that is the union of a portion of the graph $y = \sin(1/x)$, the line segment $[-1,1]$ in the ...
2
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0answers
297 views

Homology of nonorientable surfaces

Let $N_g$ be a closed nonorientable surface of genus $g$. I will try to compute the homology groups and I want you to help me with certain steps and correct my mistakes - I will use this as an ...
2
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0answers
35 views

Rational Elienberg-Maclane Spaces

Is it true that $$ H^k(K(\mathbb{Z},n);\mathbb{Q})\cong H^k(K(\mathbb{Q},n);\mathbb{Q}) $$ for all $k$?
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56 views

Seemingly serious problem with Deformation Retraction

So a friend and I are arguing over Deformation retraction. Any help to settle this would be nice. Consider a T shaped subspace of $\mathbb{R}^2$. Let $A$ be the vertical segment and let $B$ be the ...
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0answers
38 views

fibre bundle on [a,b]. prove that every fibre bundle on it is trivial.

prove that every fiber bundle on [a,b] is trivial. Please prove this elaborately. Is the Lebesgue covering lemma is required to prove this result?
2
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1answer
64 views

Collapse of a subspace - Cofibration

Let $i:A \rightarrow X$ be a (closed ) cofibration (i.e a cofibration in the Strøm Model structure). For a subspace $B \subset A \subset X$, when is it true that $A/B \rightarrow X / B$ is a ...
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1answer
134 views

Good book for studying $S_\infty$.

I'm looking for any books with some good information involving $S_\infty$ and other Polish groups. Specifically interested in $S_\infty$. This is an extremely amazing topological group, now having ...
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2answers
405 views

The degree of antipodal map, composition of reflections?

Here is a bit from Hatcher's book: I don't understand part (f); why is the antipodal map the composition of $n+1$ reflections? Even if I accept that, I still don't know why does it have degree ...
2
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2answers
256 views

When to attach 2-cells in Cayley complexes?

In Hatcher's Algebraic Topology section 1.3, Cayley complexes are explained. The book states that we get a Cayley complex out of a Cayley graph by attaching a 2-cell to each loop. There is an example ...
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2answers
44 views

A nonseparation theorem for arcs on $S^{2}$

Let $A$ be a simple unclosed curve in $S^{2}$. Is there a simple way to prove that $S^{2}\setminus A$ is path-connected using homology? By simple unclosed curve I mean that $A:I\rightarrow S^{2}$ is ...
3
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1answer
269 views

Cohomology groups of a homotopy fiber

I am reading the following: http://www.indiana.edu/~jfdavis/teaching/m623/book.pdf and on page 316 there is a thing that gets me confused: Consider the following situation: Assume that we know that ...
2
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1answer
177 views

Formula for Euler characteristic for quotient space of a CW complex

I know that there is formula for Euler characteristic: $$\chi(A\cup B)=\chi(A)+\chi(B)-\chi(A\cap B)$$ Is there any formula that links between (for CW complex) some complex, subcomplex and quotient ...
2
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1answer
126 views

Rick Miranda exercise complete intersection curve. Prove it and find genus.

The book by Rick Miranda asks to prove that the curve in $\mathbb{P}^3$ defined by the two equations $x_0x_3=2x_1x_2$ and $x_0^2+x_1^2+x_2^2+x_3^2=0$ is a smooth complete intersection curve. Also asks ...
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0answers
46 views

Build a moduli space using homotopy theory.

I'd like to build ''with hands'' the moduli space of stable principal $U(1)$-bundles over a Riemann surface $M$ of genus $2$. I have the follow idea: We know that the isomorphirsm classes of principal ...
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1answer
474 views

Homeomorphism between complex torus and S1 x S1

We have the lattice L={$m_1w_1 + m_2w_2 | m1,m2 ∈ \mathbb Z, w1,w2 ∈ \mathbb C $}. We want to construct an homeomorphism between $\mathbb C/L$ and $\mathbb S^1 \times \mathbb S^1$. I've read that the ...
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1answer
69 views

Topology question??

Suppose we have a surface $S$. Also, suppose we remove $2$ discs from the surface $S$ and we glue the boundary circles of these two discs together. Is the result a surface?? My believe is that it is ...
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1answer
34 views

Homeomorphism preserving partitions

Let $X$ and $Y$ are homeomorphic topological spaces. Consider a equivalent relation $R_X$ and $R_Y$ that partition $X$ into $X_1,\ldots,X_n$ of and $Y$ into $Y_1,\ldots,Y_n$ respectively. $X_i$ is ...
4
votes
2answers
376 views

HNN extensions as fundamental groups

I have heard that the Seifert–van Kampen theorem allows us to view HNN extensions as fundamental groups of suitably constructed spaces. I can understand the analogous statement for amalgamated free ...
1
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1answer
49 views

Why is this map continuous? (Cofibrations)

Assume that we have a map $i : A \rightarrow X$ which is a (closed) cofibration and a homotopy equivalence. Then, $A$ is a strong deformation retract of $X$ and there is a function $u: X \to I$ such ...
3
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1answer
307 views

Homology of some quotient of $S^2$

Let $X$ be the quotient space of $S^2$ under the identifications $x\sim-x$ for $x$ in the equator $S^1$. I want to compute the homology groups $H_n(X)$. I've seen this but didn't really understand. ...
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1answer
97 views

Help with Cohomologies and Homologies

My algebraic skills are very weak, so in answering please assume I know close to nothing about algebra, geometry, forms, or the like. I am trying to compute homologies and cohomologies. For ...
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1answer
538 views

Find the fundamental group of torus with two points removed [duplicate]

I'm trying to find a fundamental group of Torus \ {two points}. Any help would be really appreciated.
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0answers
242 views

Want to show two maps are homotopic

I am trying to solve the following problem but so far I cannot do it. Let $X$ be a connected CW-space such that its homotopy group is 0 except for the fundamental group. Let $M$ be a closed manifold ...
3
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0answers
28 views

Relative Hopf degree theorem

If $f,g$ are two maps from $(D^n,S^{n-1})$ to $(D^n,S^{n-1})$ such that they have the same degree, that is $f_*[\mu]=g_*[\mu]$ where $[\mu]$ is a generator of $H_n(D^n,S^{n-1})$, then can we find a ...
2
votes
1answer
84 views

Directly showing $H_0(X, x_0) \cong \widetilde{H}_0(X)$?

I'm trying to show directly that $$ H_0(X, x_0) \cong \widetilde{H}_0(X) $$ where $x_0$ is a point in the topological space $X$ and $\widetilde{H}_0$ denotes the zeroth reduced (singular) homology ...
4
votes
1answer
507 views

Intuition of Chern-Weil theory

Let $ P \rightarrow M$ be a $G$-principal bundle. The lie algebra of $G$ is $\frak{g}$ and $P$ has connection form $\omega \in H^1(P,\frak{g})$ and curvature form $\Omega \in H^2(P,\frak{g})$. We ...
6
votes
1answer
703 views

Long exact sequence for cohomology with compact supports

Related to my previous question here. Let $X$ be a topological space and let $H_c^{\bullet}(X)$ denote its singular cohomology with compact supports (rational coefficients). Let $U$ be an open subset ...
3
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1answer
848 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 ...
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0answers
72 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
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1answer
48 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.
2
votes
1answer
167 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
267 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 ...
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3answers
784 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 ...
4
votes
1answer
1k 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 ...
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1answer
44 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
268 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
110 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
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2answers
134 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 ...
2
votes
1answer
58 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 ...
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1answer
51 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
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0answers
123 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 ...
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1answer
52 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 ...
4
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2answers
267 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 ...
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1answer
99 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
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1answer
276 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 ...
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0answers
89 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 ...
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0answers
92 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 ...
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0answers
153 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 ...