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

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How do we view natural transformations as functions

1.The definition asserts that natural transformation is a map of two functors. However, from the definition, given tow functors $F,G:C,D$, we associate every element $x$ in $Obj(C)$ a morphism $F(x) ...
3
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1answer
22 views

Property of second Steifel-Whitney class of tangent bundle?

Let $M$ be a closed, smooth, simply-connected $4$-manifold. Is $w_2$ the unique class in $H^2(M, \mathbb{Z}_2)$ such that $w_2 \cup x = x \cup x$ for all $x \in H^2(M, \mathbb{Z}_2)$ or not? I ...
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1answer
15 views

Is the transfer homomorphism in cohomology surjective?

Let $p:\widetilde{X} \to X$ be an $n$-sheeted covering space. Consider a singular simplex $c:\triangle^k \to X$, because the simplex is simply-connected, there exist $n$ different lifts ...
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1answer
15 views

Homotopy $I^2 \rightarrow S^1$ lifting lemma proof

In case of a homotopy $h: I^2 \rightarrow S^1$ we can define lifting as such an $\tilde{h}: I^2 \rightarrow \mathbb{R}$ that $e^{i\tilde{h}}=h$. The existence of $\tilde{h}$ requires a proof. A way to ...
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1answer
8 views

Regarding embeddings and homological/cohomological injectivity

Possibly low-level question here: if one has an embedding $M\hookrightarrow X$, call it $\iota$, then this is of course an immersion, so the differential maps $\iota_{\ast}:T_{p}M\rightarrow ...
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1answer
15 views

Spaces homotopy equivalent to finite CW complexes

I'm doing a project about Topological Complexity (it doesn't matter what it is for the questions I will ask) and I have proofs for a few results about the bounds of the topological complexity of ...
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1answer
27 views

Prove closed disc $D^n$ is homeomorphic to the cone $CS^{n-1}$

I need to find a continuous surjective map from $D^n$ to $CS^{n-1}$. For 2 dimensions, we can use $$f: S^1 \times I /S^{1} \times \{1\} \rightarrow D^2$$ with $f(\theta,t) = (1-t)e^{i \theta}$ ...
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1answer
23 views

Is torus w. disc removed homotopic to klein bottle w. disc removed?

I know that homeomorhic spaces are homotopic, but am not sure if this applies, since I think they are not homeomorphic due to orientability. I know f and g are homotopic if they represent: ...
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1answer
17 views

strange implication of addivity axiom in homology theory

Let $H$ be a homology theory satisfying Eilenberg-Steenrod axioms and $X$ an arbitrary topological space. We can write $X$ as a disjoint union of its points $$X= \coprod_{x \in X}{\{x\}}$$ Now the ...
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0answers
9 views

Holonomic splitting

I am reading the book "Introduction to the h-Principle" by Eliashberg and Mishachev. At the moment I try to understand the Section 1.7 Holonomic splitting on page 12 but without success. I do not ...
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1answer
29 views

Fundamental Property of Regular CW Complexes

$\newcommand{\R}{\mathbf R}$ For a cell $e$ in a CW complex, we write $\partial e$ to denote $\bar e-e$. Note that $\partial e$ may not be the topological boundary of $e$ in $X$. A CW complex ...
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0answers
22 views

multiplication in H-space and loopspace of the H-space

Let $X$ be an $H-space$, and let a multiplication $,\cdot,$ be given, associative up to homotopy. Let $\Omega X$ be the loopspace of $X$ based at the identity and let the multiplication $ \circ $ on ...
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3answers
48 views

Examples of global properties that don't arise from local knowledge

Let $M$ be a smooth manifold. As an example of a global property that arises from local data we know that if $(M,g)$ is a compact surface without boundary then the Euler characteristic is given by $$ ...
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0answers
30 views

Using transfinite induction to compute fundamental groups

I want to compute the fundamental group of a space containing comb space along with boundary of $I \times I$ where I is $[0,1]$. Here is my attempt using transfinite induction and Van Kampen. Let's ...
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0answers
22 views

Can we transform a non-simple arc on a torus to a simple one?

Let $T^2$ be a torus. Assume $\gamma_1$ and $\gamma_2$ be two basis generators of $H_1(T^2)$ such that $\gamma_1$ is a circle along the meridian and $\gamma_2$ is a circle along the longitude of the ...
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1answer
25 views

Long exact sequence of a triple: working through the geometry

Suppose $X$ is a topological space with subspaces $X \supset U \supset A$ such that $U$ deformation retracts onto $A$. We know that $H^*(X,U) \cong H^*(X,A)$--one way to see this is to take the long ...
2
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1answer
24 views

Order of element in polynomial ring in Hatcher

So I've been reading Hatcher and I am unsure what they mean when they say things like $H^\ast(\mathbb{R}P^n;\mathbb{Z}_2)\cong \mathbb{Z}_2[\alpha]/(\alpha^{n+1})$ where $|\alpha|=1$. It is this last ...
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1answer
16 views

Induced map in cohomology of a covering [on hold]

Is it true that if $p: E \to B$ is a $2$-fold covering, the map $p^*$ induced in cohomology is surjective?
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1answer
52 views

What kinds of transformations preserve network topology?

I have been reading a number of "network science" papers where the authors perform transformations on networks that seem to preserve the topology of those networks. By "topology", I mean a collection ...
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1answer
37 views

Are the space of paths with two given endpoints in a contractible space, contractible?

This question is inspired by an answer to Nitrogen's answer to my Are the path connected components of $\Omega S_1$ contractible? . Here we are asked whether the space of paths in $\mathbb{R}$ ...
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0answers
32 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 $\tilde{f}:\tilde{M}\rightarrow \tilde{M}$ ...
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1answer
19 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 ...
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1answer
48 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) ...
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1answer
32 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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1answer
33 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 ...
2
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1answer
23 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 ...
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2answers
68 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 ...
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1answer
46 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 $$ H^2(M;\mathbb{Z})=\mathbb{Z}_{2k}, \text{ } k\geq 3; $$ $$ H^3(M;\mathbb{Z})=\mathbb{Z}\times\mathbb{Z}_{2}; $$ $$ ...
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0answers
22 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, ...
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1answer
39 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 ...
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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 ...
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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 ...
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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 ...
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 ...
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1answer
38 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}) ...
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1answer
42 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 ...
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1answer
44 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
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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 $H^*(S^1\times X,\{\text{pt}\}\times X;R)\cong H^*(D^1\times X,\partial D^1\times X;R)$ ...
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0answers
30 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 ...
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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
66 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 ...
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3answers
62 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. ...
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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 ...
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0answers
35 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 ...
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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 $\widetilde{H}_0(X;\mathbb{Z})\cong \mathbb{Z}$ ...
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0answers
24 views

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

Where I can find a complete description of the vector bundles of $\mathbb{P}^2$?
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1answer
38 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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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 ...
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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 ...
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1answer
71 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: ...