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

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2
votes
1answer
113 views

Restriction of sections a weak homotopy equivalence?

Suppose we are given a fiber bundle $p:E \to B$ and a point $x \in B$. Denote by $p \big|_{p^{-1}(x)}:p^{-1}(x) \to B$ the restricted fiber bundle and by $\Gamma^0(p)$ (resp. $\Gamma^0(p ...
1
vote
1answer
239 views

A question about the proof that a homotopy equivalence induces an isomorphism

If $f:X \rightarrow Y$ is a homotopy equivalence with homotopy inverse $g$ and I want to prove that $f_*$ is an isomorphism then Hatcher (on page 37) uses the following: $$ \pi_1 (X, x_0) ...
6
votes
1answer
338 views

generalization of the Jordan curve theorem

Here is a plausible generalization of Jordan curve theorem which I couldn't find a rigorous proof for it. Let $K$ be a compact subset of $\mathbb{R}^2$ which is homotopic equivalent to $S^1.$ Prove ...
5
votes
0answers
211 views

Smith normal form of graded modules. (Major edit)

Ok, this is a major rewriting of my previous entry which no one answered. Let us have two graded $F[t]$-modules M and N with bases $m_1, \ldots, m_m$ and $n_1, \ldots, n_n$, respectively, and $F$ is ...
27
votes
3answers
2k views

Consequences of Degree Theory

I'm preparing a presentation on an overview of algebraic and differential topology, and my introduction includes some motivational material on Degree Theory. I have two fundamental and invaluable ...
3
votes
1answer
271 views

Homotopic maps induce isomorphic pullbacks of a principal bundle. How functorial can this be?

More precisely, I'm trying to show that the groupoid $\mathscr{B}G(X)$ of principal $G$-bundles over $X$ and isomorphisms is equivalent to $\Pi_1(BG^X)$. It seems like the right direction to try to ...
4
votes
1answer
1k views

Proof: A loop is null homotopic iff it can be extended to a function of the disk

I would like to prove the following basic fact related to the fundamental group: A loop $\gamma : [0,1] \rightarrow X$ is null homotopic if and only if it can be extended to a continuous function of ...
6
votes
1answer
672 views

If a manifold M has zero Euler characteristic, there is a non-vanishing vector field on it

There is hint: if M has isolated singular points, find a diffeomorphism to make these singular points in a any neighborhood which you want. How can we do next?
3
votes
1answer
749 views

$M$ is a compact manifold with boundary $N$,then $M$ can't retract onto $N$.

There is hint: Prove $H^{n-1}(N) \to H^{n-1}(M)$ is trivial. Just don't know how to prove this.
7
votes
2answers
683 views

The product of sphere and torus is parallelizable. How to prove this?

The product of sphere and torus is parallelizable. How to prove this? Help solve this question, this is qualifying exams of Hopkins university.
3
votes
0answers
120 views

What is the range of the fundamental group? [duplicate]

Possible Duplicate: is the group of rational numbers the fundamental group of some space? Given any group G,can we find a complex X whose fundamental group is G.If not,which kind of group ...
8
votes
1answer
349 views

Cones on surfaces

I found this question in one of the exams given for a topology course and I couldn't get anything out of it; it just seemed overwhelming as a question, but maybe I'm missing something. Let us define ...
2
votes
1answer
317 views

Meaning of “topologically equivalent”

I've been wondering about "topologically equivalent" for some time now. For example: $S^1$ is "topologically equivalent" to $\mathbb{R}P^1$. I see that they are homotopy equivalent. But are they ...
9
votes
1answer
4k views

Homology of surface of genus $g$

This is a homework question given to me by someone of the community here and it's a generalisation of this. I was wondering if you could have a look and tell me if it's right. Thanks for your help! ...
8
votes
3answers
2k views

Homeomorphism of the Disk

I'm working through Massey's "Basic Course in AT." One of the problems is prove that a homeomorphism of the closed disk maps the boundary to the boundary and the interior to the interior. How would ...
31
votes
2answers
3k views

Which manifolds are parallelizable?

Recall that a manifold $M$ of dimension $n$ is parallelizable if there are $n$ vector fields that form a basis of the tangent space $T_x M$ at every point $x \in M$. This is equivalent to the tangent ...
7
votes
1answer
624 views

Question on the symmetric product $\mathrm{Sym}^g\Sigma$

Let $\Sigma$ be a genus $g$ closed Riemann surface. Recall that the symmetric product $\mathrm{Sym}^g\Sigma$ is the quotient of the $g$-fold product $\Sigma \times \dots \times \Sigma$ under the ...
2
votes
1answer
412 views

Fundamental Group

Consider some space $X$. Does the fundamental group tell us information about the equivalence between two paths $f,g: I \to X$? So there exists a homotopy $h: I \times I \to X$ such that $h(s,0) = ...
8
votes
1answer
935 views

What's the loop space of a circle?

Is it true that the loop space of a circle is contractible? Consider the long exact sequence in homotopy for the path fibration $\Omega S^1 \rightarrow \ast \rightarrow S^1$ shows all homotopy groups ...
2
votes
1answer
95 views

G-complexes and regular covering

Suppose $X$ a free $G$-complex (i.e. a CW-complex with a free $G$-action that permutes the cells). I would like to show that the projection $$X\overset{p}{\to}X/G$$ is a regular covering spaces with ...
6
votes
1answer
720 views

Linking Number and Intersection Number

Let $\Sigma$ be a smooth compact surface in $\mathbb{R}^3$ (for simplicity). If a closed curve $\gamma$ is linked with $\partial \Sigma$ with linking number $n$ (mod 2) then $\gamma$ should ...
4
votes
3answers
666 views

Group structure on Eilenberg-MacLane spaces

How do we put a group structure on $K(G,n)$ that makes it a topological group? I know that $\Omega K(G,n+1)=K(G,n)$ and since we have a product of loops this makes $K(G,n)$ into a H-space. But what ...
4
votes
1answer
57 views

Topological condition for a group to be of type FL

We say that a group $G$ is of type $FL$ if there exists a resolution $L_\bullet \to \mathbb{Z}$ of finite length of finitely generated, free $\mathbb{Z}G$-modules. Now, un unproven proposition in ...
5
votes
2answers
241 views

Excision via simplicial sets

Let $X$ be a topological space, covered by a collection of open sets $\{U_\alpha\}$ (or more generally a cover by subspaces whose interiors cover $X$). Consider the singular simplicial set ...
7
votes
2answers
803 views

Cohomology ring of a product

I am trying to calculate $H^*(\mathbb{R}P^3 \times \mathbb{C}P^5,\mathbb{Z})$ as a cohomology ring. I know that $$H^*(\mathbb{R}P^3,\mathbb{Z}) = \frac{\mathbb{Z}[\alpha,\beta]}{(2 \alpha, ...
6
votes
3answers
582 views

how to compute the Euler characters of a Grassmannian?

Let $G(n,m)$ be the Grassmannian of all n-dim subspaces of an m-dim vector space over $\mathbb{C}$. How to compute the Euler characters of $G(n,m)$? For example, $G(1, 2)$ is $\mathbb{C}P^1$ which is ...
3
votes
1answer
190 views

Surgery diagram and fundamental group

Let $Y^3$ be a 3-manifold obtained by surgery on $S^3$ along hopf-link with framing $p,q\in \mathbb{Z}$. I know that $Y^3\cong L(pq-1,p)$ from the Rolfsen twist. But, I wonder how can I compute the ...
1
vote
1answer
185 views

Ring structure on K-theory of an even-dimensioned sphere

I am slightly confused by some statements in Hatcher's Vector Bundle book (page 60). To start with (and I am happy with) the natural ring homomorphism $$K(S^2) \simeq \mathbb{Z}[H]/(H-1)^2$$ I am ...
8
votes
1answer
1k views

Topological Join and Wedge Sum of Spheres

Let $S^{n}$ be an $n$-sphere. I'd like to compute the reduced homology (with $\mathbb{Z}$-coefficients) of the space $\bigvee^{r}_{i = 1} S^{n_{i}} * \bigvee^{s}_{j = 1} S^{m_{j}}$, where $r, s, n_i, ...
4
votes
3answers
491 views

Calculation of Ext

Let $A$ be an abelian group. I know that $Ext_\mathbb{Z}^1(\mathbb{Z}/p,A)=A/pA$. Are there any similar formula about $Ext_\mathbb{Z}^1(A,\mathbb{Z}/p)$? I know that $Ext_R^n(A,B)\neq Ext_R^n(B,A)$ ...
2
votes
0answers
266 views

fundamental group of closed surfaces as CW complexes

Let $T$ denote the $2$-torus, $P$ the projective plane, and $nT$/$nP$ the connected sum of $n$ tori/$n$ projective planes respectively. 1) how can I prove, that $nT$ and $nP$ are homeomorphic to ...
4
votes
1answer
151 views

Ring structure of K-theory of a wedge of spheres

I've just been using Bott Periodicity to calculate the K-theory of some simple spaces - spheres, torus, and wedge of spheres. The wedge of spheres is interesting. Given that $$\tilde{K}(X \vee Y) = ...
1
vote
1answer
87 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
votes
1answer
494 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 $\rho : G\times X\rightarrow X$ ...
3
votes
1answer
129 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
vote
2answers
128 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
vote
0answers
91 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
votes
1answer
283 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
votes
1answer
132 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
votes
0answers
224 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
vote
1answer
292 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 ...
5
votes
2answers
398 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
votes
1answer
421 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
842 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' ...
6
votes
3answers
245 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 ...
4
votes
2answers
598 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 ...
10
votes
1answer
412 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 ...
5
votes
1answer
307 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
vote
0answers
441 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
433 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 $\pi_{1}(\mathbb R^{3}-X)$ ?