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

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1
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1answer
80 views

Induced map between a space and an Eilenberg Maclane space

Why does there exist a map $X\rightarrow K(H_i(X;\mathbb Q),i)$ corresponding by the universal coefficient theorem to $H_i(X;\mathbb Z)\rightarrow H_i(X;\mathbb Q)$ induced from the inclusion ...
21
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1answer
468 views

Shrinking Group Actions

Suppose $H\subset G$ is a subgroup of a topological group $G$, and $Y\subset X$ is a subspace of a topological space $X$. Suppose we are given a continuous group action $\rho : G\times X\rightarrow X$ ...
2
votes
1answer
105 views

Minimal number of contractible sets covering $\mathbb{CP}^3$

In an exam recently, I was asked to find the minimal number of contractible sets covering $\mathbb{CP}^3$ by considering the cup-product on relative cohomology. Is there nice a way of doing this, ...
1
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2answers
123 views

homology functor from abelian groups to abelian groups

given a topological space $X$, $H_n(X,-)$ is a functor from the category of abelian groups to itself. i want to clarify the following : 1) given an homomorphism $f:G\rightarrow H$ of abelian ...
1
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0answers
84 views

using direct limit argument in homology

suppose we know $H_*(K(\mathbb Q,r);\mathbb Q)$ and want to determine $H_*(K(G,r);\mathbb Q)$ where $G$ is a $\mathbb Q$-vector space. if $G$ if finite dimensional then we can use $K(H_1\times ...
0
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1answer
262 views

The Hairy Ball theorem and (non-orientable) real projective plane

Is it possible to prove the Hairy Ball theorem via non-orientability of $P^2(\mathbb{R})$? That is, the non-vanishing section $s \colon S^2 \to TS^2$ would induce (via “2-to-1” bundle $p \colon S^2 ...
0
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1answer
129 views

induced isomorphism on homotopy group

let $X$ be a topological space. suppose $\pi_i(X)=\mathbb Z$. let $f:S^i\rightarrow X$ be a representative of the generator of $\pi_i(X)$. $f$ induces an homomorphism $f_*:\pi_i(S^i)\rightarrow ...
2
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0answers
205 views

Universal coefficient theorem

Let $M$ be an $R$-module where $R$ is a P.I.D. we have the exact sequence $$0\rightarrow \operatorname{Ext}_R(H_{q−1}(X;R),M) \rightarrow H^q(X;M) \rightarrow ...
1
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1answer
283 views

Is it possible to define branched covers without using orbifolds?

One may define an orbifold by Thurston's definition as a Hausdorff space $X_O$ with open cover $\{U_i\}$ such that each $U_i$ is homeomorphic to the quotient of an open set of $\mathbb{R}^n$ by a ...
4
votes
2answers
334 views

equivalent definitions of orientation

I know two definitions of an orientation of a smooth n-manifold $M$: 1) A continuous pointwise orientation for $M$. 2) A continuous choice of generators for the groups $H_n(M,M-\{x\})=\mathbb{Z}$. ...
6
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1answer
381 views

Homology of tori and the Universal Coefficient Theorem

I'm working on something that involves tori, and specifically I'm looking for the first homology group of the $n$-torus with coefficients in $\mathbb{Z}_2 = \mathbb{Z}/2\mathbb{Z}$. However, the ...
8
votes
3answers
769 views

Where do I learn the basics of cohomology?

I am fishing for a textbook on basic Algebraic Topology. Almost every where I looked, I saw praises for Hatcher's textbook. Now, I already know a little bit of Homology (at the level of Munkres' ...
5
votes
3answers
240 views

If $n\neq m$ then $\mathbb{R}^n$ is not homeomorphic to $\mathbb{R}^m$

I want to prove that if $n\neq m$ then $\mathbb{R}^n$ is not homeomorphic to $\mathbb{R}^m$. This deceptively simple topology question came up on an algebraic topology worksheet on which the rest of ...
3
votes
2answers
500 views

A little help on the homology of a torus relative to a circle

First I'll go through my working. Throughout we assume the homology groups of the torus and circle are known. Let $X=S^1 \times S^1$ be the torus, and $A=S^1 \times \{1\}$. The following is part of ...
8
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1answer
340 views

Double-cover of a Klein-bottle-esque Space

I'm trying to complete the following exercise I found in a topology book: Construct a space A which is path-connected with fundamental group equal to $\langle r,s | r^2 s^3 r^3 s^2 = 1\rangle$, and ...
2
votes
1answer
239 views

Maps between Eilenberg–MacLane spaces

I was re-reading an algebraic topology book the other day, and I came across the following problem: Suppose that $\pi$ and $\rho$ are abelian groups and $n\geq 1$. Determine ...
1
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0answers
361 views

Fundamental group of the quotient space of the disk obtained by identifying points on the boundary that are 120 degree aparts

Let $X$ be the quotient space of the disk, $\{(x,y)\in \mathbb R^{2} \ | \ x^{2}+y^{2}\leq 1 \}$, obtained by identifying points on the boundary that are $120$ degrees apart. How can we find the ...
0
votes
1answer
387 views

Fundamental group of the complement of the closed disk in $\mathbb R^{3}$

What is the fundamental group of the complement of the closed disk in $\mathbb R^{3}$ ? i.e $X = \{(x,y,z)\in \mathbb R^{3} \ | \ z=0, \ x^{2}+y^{2} \leq 1\}$ what is $\pi_{1}(\mathbb R^{3}-X)$ ?
14
votes
2answers
397 views

Why is the Long Line not a covering space for the Circle

I know of several reasons why the long line can't be a covering space for the circle, but I'm more curious in what exactly goes wrong with the following covering map. Let $L$ be the long line and ...
2
votes
2answers
187 views

$X\!\supseteq\!K\!\simeq\!0\Rightarrow X\!\simeq\!X/K$ ($\pi_1$ of a connected graph is free)

How can I prove the following: If $X\supseteq K$ is contractible, then the quotient $X/K$ is homotopy equivalent to $X$? Since $K$ is contractible, we have a homotopy $H:id_K\!\simeq\!c_{k_0}$ ...
5
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2answers
572 views

homology isomorphic to cohomology

is it true that when we compute homologies and cohomologies with coefficients in a field then homology and cohomology groups are isomorphic to eachother? That is valid when homology groups are free ...
7
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2answers
296 views

Is there a “spanning simplex” method to calculate homology of low dimensional simplicial complexes?

I had not noticed until recently a fairly easy method to compute the homology of a $1$ dimensional simplicial complex: If $Δ$ is a $1$-dimensional simplicial complex, then find a spanning forest ...
2
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2answers
293 views

$\mathbb{R}^n$ contains no subspace homeomorphic to $S^n$ (Borsuk-Ulam)

I want to prove that $\mathbb{R}^n$ contains no subspace homeomorphic to $S^n$, as per the wikipedia article I'm trying to set up a function $g:S^n \to \mathbb{R}^n$ that would be injective, which ...
8
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1answer
209 views

Homotopy and watershed

homotopy is a new word to me. Upon trying to understand this property, I immediately think of another well-known segmentation algorithm: watersheds. I see that watershed should exhibit some ...
1
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1answer
80 views

Is there a finite generating set for the Torelli group $T_2$?

D.Johnson showed in 1983 that for g>2 , the Torelli group $Tg$ has a finite set of generators. I have not been able to find out what the case is for g=1,2; does anyone know of any result for ...
0
votes
0answers
159 views

transversal intersection and Poincaré duals

If I have $A$, $B$ two submanifolds of dimension n each included in a $2n-$manifold $M$ whose n-cohomology group is free of rank 1 and generator $\alpha$ .denote $\epsilon_{A}$ and $\epsilon_{B}$ both ...
0
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0answers
130 views

homology Questions

I have some questions and would be infinitely grateful to you for your answers: 1- $f^{*}$ being the dual of $f_{*}$ so the degree (between top dimensional (co)homology groups) is the same for both ...
1
vote
1answer
194 views

degree vs determinant in the cohomology of torus

I have read that the degree induced by a linear map on a torus $T^{n}$ is its determinant. How could one prove that result? Does this rule extend to any linear map on a space X? Thanks
0
votes
1answer
122 views

defining a fibration by eilenberg maclane spaces

Consier the fibration $$\Omega K(\mathbb Q / \mathbb Z, n)\rightarrow K(\mathbb Z , n) \stackrel{f}{\rightarrow} K( \mathbb Q, n)$$ The context is that this fibration is a crucial step in the proof ...
0
votes
1answer
138 views

Need some help computing the homology of a quotient of $T^2\times[0,1]$

I have to admit that I spent a while now thinking about the question below. I could see that the map f takes integers to integers keeping thus taking the vertices of $T^{2}$ to vertices of $R/Z$. I am ...
0
votes
1answer
214 views

euler characteristic and maps' degrees

I am wondering if there is a link between euler characteristics of orientable manifolds X and Y if there is a map of degree k from X to Y. I know about the result for a n-sheet covering map. Does the ...
2
votes
0answers
94 views

If a map $f\colon S_g\to S_g$ induces the identity in $H_1(S_g,\mathbb{Z})$, then $f$ induces the identity in $H_1(S_g,\mathbb{Z}_2)$

all: This should not be too hard, but I am stuck. $S_g$ is the orientable, genus-g surface, and $H_1(S_g,\mathbb{Z})$ is the first homology with coeffs. in $\mathbb{Z}$. I am trying to avoid ...
14
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1answer
2k views

precise official definition of a cell complex and CW-complex

I would be very grateful If someone could state a precise definition (direct one and inductive one) of a cell complex and CW-complex, since my intuition is telling that some restriction is missing and ...
2
votes
1answer
286 views

Cohomology rings of (some) sphere bundles over spheres

Recall that 2-dimensional complex vector bundles over $S^4$ are classified by $\pi_4(BU(2))=\pi_4(BU)=\mathbb Z$. For any integer $\lambda$ one can consider projectivisation of the corresponding ...
1
vote
1answer
227 views

Why is this isomorphic to the reduced homology group?

I am looking at the Mayer-Vietoris sequence for the suspension $\Sigma X$ of a space $X$, defined as $X\times [-1,1]$ with the usual identifications, with subsets $A, B$ defined to be the following $$ ...
0
votes
0answers
146 views

maps and homology again

if I have a map between two p-dimensional smooth manifolds M and N. 1-is it true that if $f_{*}$ is an isomorphism from $H_{p}(X)$ to $H_{p}(Y)$ Then f induces an isomorphism from $H_{k}(X)$ to ...
2
votes
0answers
81 views

comparing Betti numbers

My question is about what one could say about the betti number of both spaces X and Y relatively to each other if we have a map f between them (eg. a classical case is when f is a covering map) is ...
2
votes
0answers
162 views

Dimensions of Homology Groups

What does a dimension of a homology group tell us? In particular, suppose we form an arbitrary simplicial complex $S(G)$ from a simple graph $G$. Then we compute the homology groups of $S(G)$ and note ...
3
votes
3answers
2k views

About maps between homology groups

When it comes to induced homomorphisms between homology groups, I have trouble understanding which are surjective or injective. For example, the unique map $p: X \to \{x\}$ where $x$ is a point in $X$ ...
12
votes
2answers
430 views

$f:\mathbb{S}^1\rightarrow\mathbb{S}^1$ odd $\Rightarrow$ $\mathrm{deg}(f)$ odd (Borsuk-Ulam theorem)

I'm having trouble understanding the proof of Borsuk-Ulam theorem ($n=2$) that we did in our class. The only problematic part is the last sentence in the proof of lemma 1. ...
6
votes
1answer
3k views

Homology groups of torus

I computed the homology groups of the torus, can someone tell me if this is correct? The calculation, not the result that is. Thanks! The cells of $T^2$ are $e^0, e^1_a, e^1_b, e^2$ The chain groups ...
1
vote
1answer
1k views

Computing the homology groups of the torus or a cell complex

I've found this table of homology groups of the tori $T^n$. My question is: How did they compute these? More generally: what's the "recipe" to compute the homology group of say, a cell complex? ...
15
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2answers
673 views

How to understand the Todd class?

I am reading the article "K-Theory and Elliptic Operators"(http://arxiv.org/abs/math/0504555), which is about Atiyah-Singer index theorem. In page 14 the article discussed the Thom isomorphism: ...
1
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1answer
140 views

cohomology of product

I shall be thankful to you for helping me understand what I have highlighted in yellow. I see that $\gamma, \alpha$ and $\beta$ are not the same as the generators of homologies but rather the ...
0
votes
0answers
106 views

Fibration $v:S^1 \to \mathbb{R}P^1$ and a nontrivial element of $\pi_1(\mathbb{R}P^n,\ast)$

Let $i:\mathbb{R}P^1 \to \mathbb{R}P^n$ be given by $[x_0,x_1] \mapsto [x_0,x_1,0,\ldots,0]$, with $n \ge 2$. Consider the fibration $v:S^1 \to \mathbb{R}P^1$ given by $(x_0,x_) \mapsto [x_0,x_1]$ (so ...
2
votes
1answer
137 views

induced isomorphisms from Gysin sequence

Consider the path fibration: $K(\mathbb Z,2r-1)\rightarrow PK(\mathbb Z,2r)\rightarrow K(\mathbb Z,2r).$ Suppose that $H^*(K(\mathbb Z,2r-1);\mathbb Q)=H^*(S^{2r-1};\mathbb Q).$ We want to show that ...
16
votes
2answers
1k views

How useless can the Mayer-Vietoris sequence be in general?

In an algebraic topology course I'm taking we are often asked to compute the homology groups of a space $X = A \cup B$ using the Mayer-Vietoris sequence, and it happens in all of the examples I've ...
5
votes
1answer
356 views

A homotopy equivalence between spaces $B\Gamma$ and $K\Gamma$ for a graph of groups on a graph $\Gamma$

In Hatcher's "Algebraic Topology" (p. 92), the space $B\Gamma$ (for a graph of groups on a graph $\Gamma$) is defined to be a collection of spaces $BG_v$ for each vertex $v$, which are connected by ...
8
votes
1answer
420 views

Nice underestimated elementary topology problem

There is a nice elementary topology problem (proposition) that is often missing from the introductory books on the topic. PROBLEM. Let $\varphi:\mathbb{S}^{1}\rightarrow\mathbb{S}^{1}$ be a ...
3
votes
1answer
204 views

thom space associated to a fibration

let $p:S\rightarrow B$ be a fibration wiht fiber a rational $r$-homology sphere $\Sigma^r$, i.e., $H_*(\Sigma^r;\mathbb Q)=H_*(S^r;\mathbb Q)$. to such a fibration we associate its Thom space ...