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

learn more… | top users | synonyms (1)

0
votes
1answer
27 views

Requirements too stringent for singleton homotopy class [X,Y]?

I recently had a problem: Show that if $X$ is contractible, and $Y$ is path-connected, show that the homotopy class $[X,Y]$ has a single element. I have been able to prove this (I think) in a fairly ...
0
votes
1answer
67 views

Cellular homology of the real projective space $\mathbb R P^n$

I've been able to calculate the cellular homology of $\mathbb R P^2$ but I'm struggling to do the same for higher dimensions. My problem is that I don't exactly see how one get to the result $d_i: ...
1
vote
1answer
338 views

Long exact sequence for a triple follows from long exact sequence for a pair?

In homology theory, the long exact sequence for a pair $(X,Y)$ is just $H(Y)\to H(X)\xrightarrow{\partial(X,Y)}H(X,Y)\to H(Y)[-1]$. The long exact sequence for a triple $(X,Y,Z)$ is $H(Y,Z)\to ...
0
votes
0answers
21 views

$H^*(X,A;R)\cong H^*(X',A';R)\; \Rightarrow H^*(X\times Y ,A\times Y;R)\cong H^*(X'\times Y,A'\times Y;R)?$

I have a quastion about product spaces in singular cohomology. I only know a formula for sinugular homology for product spaces from lecture, the universal coefficient theorem. Let $R$ be a ...
1
vote
1answer
22 views

Extend a map over a $n+1$-cell IFF $f_nϕ$ is nullhomotopic.

Prove: If the map $f_n$ is defined on the $n$-skeleton $X_n$ and you want to define it on an $n+1$-cell with attaching map $ϕ:S_n→X_n$, then you can do so if and only if $f_nϕ$ is nullhomotopic. I ...
6
votes
1answer
40 views

Is there any result on the “counting” of minimal atlas?

Take a differentiable manifold $M$. Define $\eta(M)$ as $\min\{\#\mathfrak{A} \mid \mathfrak{A} \text{ is an atlas for $M$}\}$. For example, if $M=S^n$, we have that $\eta(M)=2$, since $S^n$ is ...
9
votes
1answer
369 views

covering space of $2$-genus surface

I'm trying to build $2:1$ covering space for $2$- genus surface by $3$-genus surface. I can see that if I take a cut of $3$-genus surface in the middle (along the mid hole) I get $2$ surfaces each one ...
4
votes
1answer
517 views

Local homology group: a homeomorphism takes the boundary to the boundary

Let $X \subset \mathbb{R}^{n}$ be the subspace $\{ x_{1},...,x_{n} \mid x_{n}\geq 0 \}$, and let $Y$ is the subspace with $x_{n}=0$. Let $x\in Y$, calculate the local homology of $X$ at ...
5
votes
0answers
69 views
+50

Codimension $1$ Embedding into $\mathbb{R}^{n+1}$

I am trying to determine which homotopy types can be realized by $n$-manifolds that have codimension one embeddings into $\mathbb{R}^{n+1}$. Suppose I have $X^n \subset \mathbb{R}^{n+1}$ and an ...
1
vote
0answers
31 views

A fibre bundle of n-tuples of vectors

The question is: does the surjective, multilinear map $\pi: (\mathbb R^\infty)^n\to \Lambda^n(\mathbb R^\infty)$ defined by $(v_1,\cdots,v_n)\mapsto v_1\wedge \cdots \wedge v_n,$ define a fibre ...
5
votes
1answer
32 views

Is $[N]^\#([N])$ congruent to $w_n(\nu_N)([N])$ mod $2$, where $\nu_N$ is the normal bundle of the embedding of $N$ in $M$?

Let $M$ be a closed, smooth, orientable $2n$-manifold, and let $N$ be a closed, smooth, orientable $n$-submanifold. Let $[N]^\#$ denote the cohomology class (Poincaré) dual to the homology class ...
0
votes
1answer
33 views

If induced map on homology is surjective, is induced map on cohomology injective?

Suppose I have topological spaces $X, Y$ and a continuous map $f: X \to Y$. Let $\mathbb{k}$ be a field, and $i \ge 1$ an integer. If the induced linear map on homology $f_* : H_i ( X, \mathbb{k}) ...
3
votes
3answers
60 views

Example of Topological Group

I'm trying to learn about topological groups but I can't seem to google anything that provides a simple and clear example of how the set of elements of a group correspond to open sets of a topology. ...
1
vote
1answer
38 views

Why is $H^*(S^1\times X,\{\text{pt}\}\times X;R)\cong H^*(D^1\times X,\partial D^1\times X;R)$?

Let $X$ be a topological space, $R$ is a commutative, unital ring. In a proof from lecture there is claimed that $H^*(S^1\times X,\{\text{pt}\}\times X;R)\cong H^*(D^1\times X,\partial D^1\times X;R)$ ...
1
vote
1answer
37 views

$T^2-D$ does not retract to the boundary $\partial D$

First of all: yes, there is already a post about it, but I missread retract as strong deformation retract and wanted to know if this solution is right if we really do assume the stronger assumption of ...
1
vote
1answer
37 views

Bott and Tu construction of chern classes

To quote from Differential Forms in Algebraic Topology, Set $x=c_1(S^*)$. Then $x$ is a cohomology class in $H^2(P(E))$. Since the restriction of the universal subbundle $S$ on $P(E)$ to a fiber ...
0
votes
0answers
22 views

Gluing along infinitely many trivial cofibrations

I am in a situation where I have a space Y obtained from a space X by gluing on infinitely many trivial cells. That is, I have a collection of maps $f_\alpha: A_\alpha \to X$ each of which is a ...
0
votes
0answers
63 views

When is a diffeomorphism analytic?

I've read somewhere "a class $C^\infty$ diffeomorphism is said to be analytic" but I forgot to write down where I read this and now I can not find it, which makes me wonder if it's true? I'm working ...
4
votes
1answer
285 views

Stiefel-Whitney numbers for product bundle

I'm reading Milnor's "characteristic classes" and I want to compute Stiefel-Whitney numbers of $ P^2 \times P^2 $ (product of projective spaces) for one of the problems, I know how Stiefel-Whitney ...
15
votes
1answer
443 views

Can we generalize the regular value theorem even beyond the Ehresmann's theorem?

The formulation is complicated, but the answer may be some clever usage of the partition of unity, because locally the answer is given by the regular value theorem and the whole problem is to glue it ...
27
votes
4answers
2k views

Why is the Jordan Curve Theorem not “obvious”?

I am horribly confused about Jordan's Curve Theorem (henceforth JCT). Could you give me some reason why should the validity of this theorem be in doubt? I mean for anyone who trusts the eye theorem is ...
1
vote
0answers
19 views

intution behind homology [duplicate]

i am currently studying a course in homology theory and have done a basic introductory course in algebraic topology which deals with the idea of the fundamental groups and their topological ...
1
vote
1answer
20 views

Do these non-homotopic maps induce the same map in reduced homology?

Consider two maps $f, g: X\to Y$, where $X=Y=\{ 0, 1 \}$ with discrete topology, $f$ is the identity and $g$ maps everything to 0. Then it's clear that $\widetilde{H}_0(X;\mathbb{Z})\cong \mathbb{Z}$ ...
2
votes
0answers
29 views

Poincaré duality isomorphism maps cohomology to homology here?

Let $M = M^n$ and $A = A^p$ be compact oriented manifolds with smooth embedding $i: M \to A$. Let $k = p - n$. Does the Poincaré duality isomorphism$$\bigcap \mu_A: H^k(A) \to H_n(A)$$map the ...
5
votes
2answers
305 views

Does the hairy ball theorem follow from Borsuk-Ulam?

The proofs I have seen for the hairy ball theorem all use either degree of a map defined in e.g. by homology or direct computations using stereographic projections in order to use homotopy arguments ...
0
votes
0answers
24 views

vector bundles of $\mathbb{P}^2$ [on hold]

Where I can find a complete description of the vector bundles of $\mathbb{P}^2$?
0
votes
1answer
37 views

Postnikov tower of a product

Let $X$ and $Y$ be simply connected, locally finite CW-complexes and let $(X_i)_i$ and $(Y_i)_i$ be their Postnikov towers respectively. Is the Postnikov tower of $X\times Y$ given by the products ...
-1
votes
0answers
89 views

Why every fundamental group isn't a trivial fundamental group?

I don't understand exactly the definition of the Fundamental Group. Munkres defined Fundamental Group in your book 'Topology' like "Let $X$ be a space; let $x_0$ be a point of $X$. A path in $X$ that ...
2
votes
1answer
54 views

Why is even codimension necessary to apply excision for the Euler characteristic?

In this answer on MathOverflow, it is claimed that $$\chi(X/Z)=\chi(X)-\chi(Z)$$ holds for complex subvarieties $Z$ only because $Z$ has even codimension. It is implied that for $Z$ with odd ...
0
votes
1answer
101 views

How should I prove the following? Algebraic topology and homeomorphism

I am struggling immensely with topology since the start of the course, probably due to its extremity; the explanations are either "very rough" or "very strict and rigid and hard to comprehend." Either ...
0
votes
1answer
88 views

What does the notation “*” mean?

I do not know the name of or what it does so I have no means of searching for an answer over the internet or a book. In my notes for algebraic topology, I have this bit that says, For any $f: X ...
1
vote
1answer
47 views

Example of Spherical Element (Simplicial Homotopy)

Definition: An element $x\in X_n$ is said to be spherical if $d_i x=*$ for all $0\leq i\leq n$. $X$ is a pointed fibrant simplicial set. I am puzzling over this definition. For instance, if ...
6
votes
0answers
183 views
+50

Hatcher 2.1.10…

Hatcher asked a question in chapter $2$ (a) Show the quotient space of a finite collection of disjoint $2$-simplices obtained by identifying pairs of edges is always a surface,locally homeomorphic ...
1
vote
1answer
36 views

A remark about the map $\partial^*:H^{*}(A;V)\to H^{*+1}(X,A)$ of the l.e.S. in cohomology

My question is about a remark from lecture about the connecting-homomorphism of the long exact sequence in homology of a pair $(X,A)$. Let $(X,A)$ a pair of topological spaces, $V$ be an abelian ...
4
votes
1answer
759 views

Does the homology, homotopy, and geometric realization functors of a simplicial group preserve colimits?

Given a based simplicial group, you can find its reduced homology with coefficients in a field, homotopy, and geometric realization. These are functors. If I have a free product of based simplicial ...
0
votes
0answers
32 views

$K(G,1)$ for a torsion-free group $G$

It is known that if a finite CW complex $X$ is a $K(G,1)$, then the group $G$ must be torsion-free. see proposition 2.45 of Hatcher Now my question is If $G$ is a torsion-free group, then is ...
0
votes
0answers
29 views

Existence of a curve with index 1 around a compact set

Let $K \subset \mathbb{C}$ be compact. If $U$ is an open set containing $K$, I want to show that there exists a collection of (piecewise $C^1$) curves $\gamma_1...\gamma_n$ such that 1) For $ x \in ...
2
votes
0answers
23 views

Cohomology of geometric realization of a simplicial topological space

Let $X$ be a simplicial topological space. We can consider to notions of cohomology of $X$. Denote by $|X|$ geometric realization of $X$. Then we can take just $H^*(|X| )$ (i.e. usual cohomology of ...
1
vote
1answer
46 views

$\mathbb CP^1 \approx S^2$ proof check

I wanted to give a whole proof of this fact as I was not able to find a detailed one myself. I have the feeling that such a proof has been asked quite frequently by several users and I hope this may ...
0
votes
0answers
8 views

Filtration on simplicial complex as a grayscale image

If we make a filtration on the complex built from vertices and edges and faces(2dimensions) of a grayscale image according to grayscale value. Does it considered as a filtration on simplicial complex? ...
4
votes
0answers
43 views

Characterisation of Stable Cohomological Operations: $\Sigma (\tau_n (\imath_{A,n}))=\rho_{n+1}^*(\tau_{n+1}(\imath_{A,n+1}))$

I've started studying (stable) cohomological operations on my lecture notes, and I was given that an equivalent definition for a family of cohomological operations to be a stable cohomological ...
4
votes
1answer
50 views

Homotopy equivalence between $X=\{0\}\cup \{\frac{1}{n},n\in \mathbb{N}\}$ and a discrete space

Consider the space $X=\{0\}\cup \{\frac{1}{n},n\in \mathbb{N}\}$ with the topology induced by the real line. Is $X$ homotopy equivalent to some enumerable discrete space $Y$? My try was the ...
0
votes
1answer
43 views

Proof check/ suggestion: The suspension of $S^n$

In one of my excercise sheets there was a remark saying that $$SX \approx S^{n+1}$$ where $SX$ denotes the suspension of $X=S^n$. So I tried to prove this on my own and would like to discuss my ...
3
votes
1answer
44 views

counterexample for $f_*:C_*\to D_*$ be a chain map such that $f_*$ induces an isomorphism in homology. Then $f_*$ is a chain homotopy equivalence

I want to understand a counterexample for: Let $f_*:C_*\to D_*$ be a chain map such that $f_*$ induces an isomorphism in homology. Then $f_*$ is a chain homotopy equivalence, because the statement ...
0
votes
0answers
24 views

How do I prove shoenflies theorem for $\mathbb{R}^2$?

I studied the contents in Munkres-Topology. In this text, the author uses basic algebraic topology to prove Jordan curve theorem. Then, he wrote that "If $C$ is a simple closed curve in $S^2$, the ...
1
vote
1answer
33 views

Degree of a non-surjective map f

In my notes I found an excercise claiming that $f: S^n \to S^n$ has $deg(f)=0$ whenever it's not surjective. I can prove this if I assume smoothness by applying Sard's theorem but I'm wondering if ...
1
vote
1answer
113 views

The closed orientable surface of genus 2

I have a very simple question to ask. What is a closed orientable surface of genus 2? Thank you in advance for helping me.
0
votes
1answer
34 views

Can a torus be cut into a Möbius strip with zero number of half twists?

It is known that the torus can be cut into a Möbius strip with an even number $n$ of half twists(half twist means rotation 180 degree). I am asking if it is possible to $n$ to be zero?
-1
votes
0answers
37 views

Topological Equivalence of Metric Spaces [closed]

Suppose we have two different metric spaces $(X,\phi)$ and $(Y,\psi)$. I need to show that the metrics $\phi$ and $\psi$ are equivalent metrics. Using a sterographic projection, I've shown that if we ...
0
votes
0answers
43 views

Compact 1-manifolds

I want to find all compact 1-manifolds. I think these will be lines, with fixed endpoints, and also the case where the endpoints meet. Hence I think all compact 1-manifolds are homeomorphic to the ...