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
0answers
4 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 ...
0
votes
1answer
24 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}) ...
5
votes
1answer
15 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 ...
1
vote
1answer
29 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
25 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 $H^*(S^1\times X,\{\text{pt}\}\times X;R)\cong H^*(D^1\times X,\partial D^1\times X;R)$ ...
0
votes
0answers
15 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 ...
1
vote
1answer
32 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
54 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 ...
2
votes
3answers
58 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
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 ...
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 ...
1
vote
1answer
18 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 $\widetilde{H}_0(X;\mathbb{Z})\cong \mathbb{Z}$ ...
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 ...
2
votes
1answer
53 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 ...
-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 ...
0
votes
0answers
42 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: ...
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 ...
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
21 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 ...
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? ...
1
vote
1answer
45 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 ...
1
vote
1answer
34 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
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 ...
3
votes
1answer
43 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 ...
4
votes
0answers
41 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 ...
0
votes
0answers
23 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 ...
0
votes
1answer
42 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 ...
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 ...
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 ...
0
votes
1answer
33 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 [on hold]

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 ...
3
votes
0answers
54 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 a embedded ...
2
votes
0answers
46 views

Cohomology algebra generated by Steifel-Whitney classes and dual classes subject to defining relations? [on hold]

Is the cohomology algebra $H^*(G_n(\mathbb{R}^{n+k}))$ over $\mathbb{Z}/2$ generated by the Steifel-Whitney classes $w_1, \dots, w_n$ of $\gamma^n$ and the dual classes $\overline{w}_1, \dots, ...
0
votes
1answer
33 views

Question about $C_f \approx \mathbb R P^2$

I'm trying to understand why the mapping cone $C_f$ (where $f: S^1 \to S^1, \space e^{2\pi it} \mapsto e^{4 \pi i t}$) is homeomorphic to the real projective space $\mathbb R P^2$. If we use the ...
0
votes
1answer
44 views

Question in the proof of the Brower fix point theorem

One can show that for any given homology theory $H$ with non-trivial coefficient group $G$ there does not exist a retract $\partial B^n \subset B^n$. Brower's fix point theorem states that any ...
1
vote
0answers
23 views

Why is the lift of a group action an action on the covering space in this case?

I am reading the covering actions section from Bredon's Transformation Groups and have the following difficulty - Let $G$ be a connected Lie group and $G^*$ be the universal covering group of $G$ ...
2
votes
1answer
39 views

How do we obtain the following identification

I don't understand geometrically why the identification below let us generate the shape on the right can someone explain or give me some intuition ?
1
vote
0answers
33 views

Question about the Hessian Criterion on a curve with singularity

So in class we have this theorem we call the Hessian Criterion: If we have a singular point in an affine curve in $\mathbb{C}^2$. Then $\frac{ \partial ^2 f}{\partial x^2}\frac{ \partial ^2 ...
3
votes
0answers
38 views

Correspondence defines embedding of $G_n(\mathbb{R}^m)$ into $G_{n+1}(\mathbb{R}^1 \oplus \mathbb{R}^m) = G_{n+1}(\mathbb{R}^{m+1})$? [closed]

Does the correspondence$$X \overset{f}{\to} \mathbb{R}^1 \oplus X$$defines an embedding of the Grassmann manifold $G_n(\mathbb{R}^m)$ into $$G_{n+1}(\mathbb{R}^1 \oplus \mathbb{R}^m) = ...
0
votes
0answers
16 views

Question Regarding Variaties finding the coresponding polynomial [closed]

Hi guys I have a general question. Say we have a set and we suspect it is a variety. How does one find the polynomial corresponding to it. I am thinking say the set ${(a^2,a^3+1)}$ where a is a ...
-1
votes
0answers
44 views

There is a theorem analogous to the Brouwer fixed-point for the 2-dimensional sphere?

Intuitively I think that for $f$: $\mathbb{S^2}\rightarrow\mathbb{S^2}$ a continuous function exist at least two fixed points. Using the same reasoning the statement of Brouwer fixed-point theorem I ...
2
votes
1answer
44 views

Given a column vector, can we add columns, continuously dependent on the given one, to get an invertible matrix?

Given a vector $x$ in the $n=6$-dimensional Euclidian space $\mathbb{R}^n$, do there exist $n-1$ continuous functions $f_1$ to $f_{n-1}$ such that the matrix $$(x,f_1(x), … ,f_{n-1}(x))$$ is ...
2
votes
1answer
38 views

How is identification is done in the definition of CW complexes

Consider the definition of CW complex from hatcher I am trying to understand the issue with the identification, because I feel there is something I don't understand. I decided to do an example and do ...
1
vote
1answer
25 views

Free group action on $S^n$ proof in Hatcher

Theorem: :$\mathbb{Z}_2$ is the only nontrivial group that can act freely on $S^n$ if $n$ is even. Proof: Since the degree of a homeomorphism must be $\pm 1$, an action of a group on $S^n$ determines ...
1
vote
0answers
47 views

Defining a continuous complex logarithm on open set $U \subset \mathbb{C}$

Suppose you are given an open set $U \subset \mathbb{C}$ and a continuous function $f: U \rightarrow \mathbb{C}-\{0\}$. And $f$ has the next property: For every closed loop $ c: I \rightarrow U$ ...
0
votes
0answers
49 views

Question about the induced Hurewicz isomorphism

In my notes it's claimed that the group homomorphism $$\Phi: \pi_{1}(X,x_{0}) \to H_{1}(X), \space \{f\} \mapsto[f]$$ clearly induces a group homomorphism $\Phi_{*}: \pi_{1}(X,x_{0})^{ab} \to ...
1
vote
0answers
51 views

Algebraic Topology problem on continuous functions over disks on sphere

I have come across the next problem, and I would like a little hint. Everything I'm thinking is not working or is a dead end. Let $f,g : D^2 \rightarrow S^2$ be continous maps such that $(x,y) \in ...