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

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0
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14 views

Lifting of a Diffeomorphism to an Orientable Double Cover

Let $M$ be a non-orientable smooth $d$-dimensional manifold. Let $\tilde{M}$ be the oriented double cover for $M$; and let $f\in Diff^{1+\beta}(M)$. Define $\tilde{f}:\tilde{M}\rightarrow \tilde{M}$ ...
6
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1answer
163 views
+50

Smash product of compact spaces

In the topology book I'm reading I found the following statement: The "smash product" (of two pointed spaces) is defined as $X \bigwedge Y=X \times Y/(X \times \lbrace*\rbrace \bigcup Y \times ...
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2answers
48 views

Non-abelian fundamental group on a path-connected space

I am doing a self-study of algebraic topology, and am having some difficulties comprehending the idea of a non-abelian fundamental group on a path connected space. (See for example Hatcher Exercise ...
1
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1answer
16 views

Definition of the Fundamental Class for $K(A,0)$

I'm having a little doubts on the definition of the fundamental class for the Eilenberg-MacLane space $K(A,0)$. Recall that a fundamental class $\imath_{A,n}$ for a polarized $K(A,n)$ is the element ...
1
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1answer
508 views

Prerequisites for Bredon's “Topology and Geometry”?

My background in topology is the first 6 chapters of Munkres's "topology" and in algebra Herstein's "Topics in Algebra". Both of them I self studied. A look at the table of contents of Bredon's ...
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0answers
31 views

Injectivity in the zero homology

I'm struggling with following step in an excercises about Mayer-Vietoris sequences: In one step the solution says this map is injective since $A \cap B$ is path-connected: $$ H_0(A \cap B) ...
3
votes
1answer
24 views

Are the path connected components of $\Omega S_1$ contractible?

Let $\Omega S_1$ be the space of loops of $S_1$ based at $x_0 \in S_1$ with the topology of uniform convergence. We know that the path connected components of this space are in one to one ...
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0answers
18 views

Long exact sequence of $(I \times Y, \partial I \times Y)$.

There's a section in chapter 3 in Hatcher's Algebraic Topology where he talks about the long exact sequence of the pair $(Y \times I, Y \times \partial I)$, where $Y$ is any topological space. The ...
3
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3answers
886 views

Holomorphic functions on a complex compact manifold are only constants

Is there a simple proof that every holomorphic function $M\to\mathbb{C}$ on a compact complex manifold $M$ is constant?
2
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2answers
141 views

Assignment - Euler characteristic constant under barycentric subdivision

I am working on an assignment and I am stuck, mostly I have no clue how to quite attack it. I don't want the answer or anything just advice on angels at which I can go about this. For an n-dimensional ...
2
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1answer
21 views

Showing $H_i(M, M - \{x \}) \cong H_i( \mathbb{R}^n, \mathbb{R}^n - \{0\} )$ via excision

I want to show $H_i(M, M - \{x \}) \cong H_i( \mathbb{R}^n, \mathbb{R}^n - \{0\} )$ via excision and can't quite figure out how to choose my subspaces. For $Z \subset A \subset X$, excision gives ...
5
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2answers
260 views

Looking for a (nonlinear) map from $n$-dimensional cube to an $n$-dimensional simplex

I am looking for a (nonlinear) map from $n$-dimensional cube to an $n$-dimensional simplex; to make it simple, assume the following figure which is showing a sample transformation for the case when ...
-1
votes
1answer
39 views

How to use the Universal Coefficient Theorem to determine $H^i(M; \mathbb{Z}_p)$ from $H^i(M; \mathbb{Z})$? [on hold]

Let $M$ be a path-connected finite $CW$-complex. Suppose $$ H^2(M;\mathbb{Z})=\mathbb{Z}_{2k}, \text{ } k\geq 3; $$ $$ H^3(M;\mathbb{Z})=\mathbb{Z}\times\mathbb{Z}_{2}; $$ $$ ...
0
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0answers
20 views

universal coefficient theorem for mod p cohomology

In the book Algebraic Topology, Allen Hatcher, p. 266, Corollary 3A.6 (b): Question: I want to rewrite the above statement into a cohomology version. If I replace all homologies with cohomologies, ...
0
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1answer
37 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
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1answer
69 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
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1answer
342 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 ...
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0answers
31 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
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1answer
25 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
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1answer
44 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
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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
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0answers
80 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
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0answers
34 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
40 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
35 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
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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
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1answer
42 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
40 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
38 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 ...
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0answers
26 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
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0answers
65 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
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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 ...
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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
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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
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0answers
33 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 ...
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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
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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 ...
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0answers
90 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
56 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
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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
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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
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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 ...
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203 views
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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
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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 ...
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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 ...