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Questions tagged [homotopy-theory]

Two functions are homotopic, if one of them can by continuously deformed to another. This gives rise to an equivalence relation. A group called homotopy group can be obtained from the equivalence classes. The simplest homotopy group is fundamental group. Homotopy groups are important invariants in algebraic topology.

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Homotopy classes of maps into the skeleta of CW complexes

Suppse $Y$ is a connected CW complex and $Y_n$ denotes the $n$-skeleton of $Y$. Suppose $X$ is a connected CW complex of dimension $n$. The inclusion $\iota \colon Y_n \to Y$ yields a map $[X,Y_n] \to ...
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Homotopy classes and CW approximation

Suppose $X$ is a connected CW complex with $\dim X = n$ and $Y$ is CW-complex which is $n$-connected. Is it true that the set of homotopy classes of maps $f \colon X \to Y$ is trivial, i.e. $[X,Y]=\{[...
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Submersion with contractible fibers

Let $f: M\to N$ be a surjective submersion of manifolds with contractible fibers. Is it true that $M$ and $N$ are homotopy equivalent?
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CW-Structure of Spin(n)

I cannot find any information about the CW-Structure of $Spin(n)$ groups. Clearly $\pi_1=\pi_2=0$ and I think $H_3(Spin(n))=\mathbb Z$ $(n\geq 5)$, so the $3$-skeleton is $S^3$. What is the $4$-...
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Does the center of $\pi_1(Y)$ act trivial on $[X,Y]_\star$?

Let $X$ and $Y$ be based (and well-pointed) and connected. We have an action of $\pi_1(Y)$ on the set $[X,Y]_\star$ of based homotopy classes of based maps. The quotient is just the set $[X,Y]$ of ...
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Do smooth, homotopic curves in a manifold have a smooth variation from one to the other?

The motivation for this is demonstrating that the proof in Simply connected manifold with nonpositive curvature has no more than one geodesic between points is correct (or learning that it is false). ...
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Is this a 'relative' colimit or some other categorical construction?

Let $\mathcal{A}$ be a closed covering of some space $X$, then let $\Sigma[\mathcal{A}]$ be the category of intersections of subsets of $\mathcal{A}$, further let $F: \Sigma[\mathcal{A}] \to \mathsf{...
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What is the type of this surface? (square with two bridges)

Take a square (without border) and build two bridges on it. You can go under the bridge or across the bridge (as my attempt at drawing it poorly attempts to describe). Since you can freely move the ...
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1answer
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Surjective homotopy equivalence which is not a fibration?

This is probably obvious to topologists so I'll just come right out with the question: What is an example of a surjective homotopy equivalence $E \to B$ of path-connected CW complexes which is not a ...
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1answer
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Relating the Nonlinear and Linear operators in the Homotopy Analysis Method

The question refers to chapter two of the book Liao, Shijun. Homotopy analysis method in nonlinear differential equations. Beijing: Higher Education Press, 2012. In section 2.3.2 of the book the ...
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Fibration in topology

Can anybody give some examples of geometrical explanation of the concept fibration in topology? I found some examples of this concept but they are mere abstract and therefore I'm unable to grasp the ...
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Finding a homotopy directly to show that two induced homomorphisms on a fundamental group are the same

Consider the following problem: “Let $I\in \pi_1(S^1,(1,0))$ be the class of the identity map. Show that $nI$ is the class of the map $f_n:S^1\rightarrow S^1$ given by $f_n(z)=z^n$. I can solve it ...
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The meaning of $\pi_1(S^1,(1,0))$

I know what $\pi_1(X,\star)$ means. But what does $\pi_1(S^1,(1,0))$ mean? I came across it when wanting to solve the following problem: “Let $I\in \pi_1(S^1,(1,0))$ be the class of the identity map. ...
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The homotopical proof of the fundamental theorem of algebra

I am reading a homotopical proof of the fundamental theorem of algebra, and the end of the proof is as follows: ... the map $z\mapsto r^nz^n$ (where $r$ is a positive real number and $z\in S^1$) is ...
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Is a left homotopy inverse of a quasi-isomorphism automatically a right homotopy inverse?

Let $R$ be an associative ring with unit. Let $E$ and $F$ be two chain complexes of $R$-modules and $\phi: E\overset{\sim}{\to} F$ be a quasi-isomorphism between them, i.e. $\phi$ induces ismorphisms ...
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Connected Hausdorff space all whose $n$-th homology groups, for $n\ge 1$, are trivial

Let $X$ be a connected Hausdorff space such that $H_n(X,\mathbb Z)=0, \forall n \ge 1$. Then does that necessarily imply that $X$ is path connected i.e. $H_0(X,\mathbb Z)=\mathbb Z$ ?
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On topological space whose homology groups/modules are trivial

Let $X$ be a simply connected topological space. Consider the following three statements : (1) $X$ is contractible. (2) For every commutative ring $R$, $H_0(X,R)=R$ and $H_n(X,R)=0, \forall n \ge 1$...
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Why homotopic equivalence is equivalent to quasi iso in K(I)

Let $X,Y \in K(A)$ where $A$ is an abelian category and $X,Y$ are complexes s.t. $X^i$ and $Y^i$ are injecive for every i. How can I prove that if $t : X \to Y$ is a quasi-isomorphism he $t$ is an ...
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Lack of homotopy equivalence between given spaces

I am trying to show that spaces $\mathbb{S}^3\times \mathbb{CP}^\infty$ and $\mathbb{S}^2$ are not homotopy equivalent. (My goal is to use then as examples of non-homotopy equivalent spaces of the ...
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Is every map from $\mathbb{R}P^2 \to S^2$ nullhomotopic?

I think the answer is no, but I'm not sure. Consider the CW-structure on $\mathbb{R}P^2$ given by one $0$-cell $x$, one oriented $1$-cell $a$ attached to $x$ at both ends, and one $2$-cell whose ...
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Algebraic topology and complex analysis

Happy new year! Can algebraic topology help me to understand deeper complex analysis ? If so, can you give me, please, an example of a book ? (the chapter that I need if it is not all the book). ...
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Initial objects on $\infty$-categories

Let $X \in \mathbf{Set}_{\Delta}$ an $\infty$-category and $\tau_1$ the left adjoint functor to the nerve $\mathrm{N} \colon \mathbf{Cat} \to \mathbf{Set}_{\Delta}$. Show that if $x$ is an initial ...
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Homotopy of continuous map from a space with finite fundamental group

Problem source: 2c on the UMD January, 2018 topology qualifying exam, seen here https://www-math.umd.edu/images/pdfs/quals/Topology/Topology-January-2018.pdf I have an argument for it, but I am not at ...
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Homotopy type of $\mathbb{R}^3 \setminus \{ \mathrm{wedge \hspace{3pt} sum \hspace{3pt} of \hspace{3pt} 2 \hspace{3pt} circles} \}$

Is there something "nice" that the space $\mathbb{R}^3 \setminus \{ \mathrm{wedge \hspace{3pt} sum \hspace{3pt} of \hspace{3pt} 2 \hspace{3pt} circles} \}$ is homotopy equivalent to? I know that $\...
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About a Kan fibration (Postnikov towers for simplicial sets)

I am trying to understand a specific construction of Postnikov towers for simplicial sets, as explained for instance here (under "absolute Postnikov tower") So you start with a simplicial set $X$ (I ...
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chain homotopy equivalence and quasi-isomorphism

Suppose $(C,d)$ and $(D,\delta)$ are two chain complexes over a field and $f:C\to D$ is a chain map. We say $f$ is a quasi-isomorphism if it induces an isomorphism of the homology groups $H(C,d)\to ...
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Understanding the homotopy operator for de Rham Cohomology

This is in John Lee's Smooth Manifold 2nd Edition, pg 444 For any smooth manifold $M$, there exists a linear map $$ h:\Omega^p(M \times I ) \rightarrow \Omega^{p-1}(M)$$ such that $$ h(dw)+d(...
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Existence of topological space which has no “square-root” but whose “cube” has a “square-root”

Does there exist a topological space $X$ such that $X \ncong Y\times Y$ for every topological space $Y$ but $$X\times X \times X \cong Z\times Z$$ for some topological space $Z$ ? Here $\cong$ means ...
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Some questions about specific topological embeddings

It often happens in algebraic topology that we have a construction on, say, a space $X$, usually obtained by taking a product and then a quotient. Then we may have a "natural" injection of $X$ in the ...
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Associativity of smash product for compact spaces [duplicate]

The following is Problem 2.2.14 in Tammo tom Dieck's Algebraic Topology: Let $Y,Z$ be compact or $X,Z$ be locally compact. Then the canonical bijection $(X\wedge Y)\wedge Z\to X\wedge(Y\wedge Z)$ ...
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If induced homomorphism of homologies of complex projective space is non zero then the map is surjective

Let $f \colon \mathbb{CP}^n \to \mathbb{CP}^n$ be a continuous function which induces a non-zero map $f_*$ on every Homology group $H_{2k} (\mathbb{CP}^n)$. Show that $f$ is surjective. As we all ...
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How does the image of the Hurewicz map $\pi_n(X,x) \to H_n(X)$ depend upon the choice of the base point?

Let $X$ be a path connected topological space. I understand that the homotopy groups $\pi_n(X,x_0)$ and $\pi_n(X,x_1)$ are isomorphic to each other. However I do not understand whether the image of ...
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Extending smooth maps from the punctured disk

Suppose that I have a smooth map of manifolds $f: M^n \rightarrow D^2\backslash 0$ and $f$ is a submersion, i.e. a fiber bundle over $D^2 \backslash 0$. When is it possible to find another smooth map ...
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How are multiplication maps of spectra defined?

In lecture 22 of Lurie's notes on chromatic homotopy theory there is the following cryptic definition. For each integer $k$, let $M(k)$ denote the cofiber of the map $Σ^{2k} \mathrm{MU}(p) \to \...
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More on flimsy spaces

I've recently encountered this very nice question on flimsy spaces and have come up with the following generalised version of the question, one which isn't also answered (in an obvious fashion) by one ...
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Show that the inclusion of the real projective plane in the complex projective plane is not null-homotopic?

In other words, how to show that $\mathbb{RP}^2$ is not contractible in $\mathbb{CP}^2$. Any hints would be appreciated.
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(wrong) Proof of stability of homotopy equivalences under pullback

In studying the category Top localized by homotopies, I asked me this question: "Is homotopy equivalence stable by pullback (base change)?" I know that it is necessary to have a further condition (...
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Is there a “low-powered” axiom system for producing all the $\infty$-groupoid axioms?

If I understand correctly, part of the buzz surrounding homotopy type theory is that a small system of axioms ends up producing all of the (weak) $\infty$-groupoid axioms, where by an "axiom" in this ...
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the fundamental group of the immersed image of Klein bottle in $\mathbb{R}^3$

how to get the fundamental group of the immersed image of Klein bottle in $\mathbb{R}^3$? I just try to use the Van-Kampen theorem to prove it is $\mathbb{Z}$. i am not sure. if u know something ...
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homotopy equivalence of these three spaces

$S^2$ with a diameter $T^2$ with a disk $S^2$ with a circle i try to find a space such that they are all the deformation contraction of it. And i failed. any idea is helpful. thanks
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Is a space homotopy dominated by a polyhedron P homotopy equivalent to a weak retract of P?

A space $A$ is homotopy dominated by a space $X$ if there are maps $f:A\to X$ and $g:X\to A$ so that $g\circ f\simeq 1_A$. Also, a subset $A$ of a space $X$ is called a weak retract of $X$ if there ...
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Weak homotopy equivalence induces isomorpism of sets of homotopy classes?

There is a question in my homework on the algebraic topology course asking if two spaces $X$ and $Y$ are weakly homotopy equivalent in case for every cellular space $Z$ sets $[Z,X]$ and $[Z,Y]$ are ...
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An alternative, more formal proof of a path lifting criterion in tom Dieck's Algebraic Topology

This is a theorem from Tammo tom Dieck's Algebraic Topology: While it has a direct proof, the author gives a more formal proof in the problems: By pullback I suppose he means a diagram $\require{...
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Kähler manifolds are formal

I want to understand why Kähler manifolds are formal. This was first proved by Deligne, Griffiths, Morgan, Sullivan Let $\mathcal M$ be a minimal differential algebra and $H^*(\mathcal M)$ the ...
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1answer
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(Co-)fibrations in Top and CGWH

Suppose that you have a map $i: A\rightarrow X$ between CGWH (compactly generated weakly Hausdorff) spaces. It is true that if it is a cofibration in CGWH (the category of CGWH spaces), then it is a ...
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Having problem with tom Dieck's algebraic topology text

(An online PDF of the text Algebraic Topology by Tammo tom Dieck can be found here.) This question is really soft. I'm having problem reading this text. Let me elaborate. I found this book too ...
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What is the theorem being mentioned here?

In this video, at just after the 5 minute mark, the speaker says: "...the colimit of the diagram has the homotopy type of the homotopy colimit. Why? Because all of the maps included in the diagram ...
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1answer
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Formality of commutative differential graded algebras

I want to understand the definition of a commutative differential graded algebra (CDGA) to be formal. Actually I encountered two definitions, but I have trouble with both. From Wikipedia: A ...
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1answer
203 views

Is every loop in a 3-manifold homotopic to some loop on its boundary?

Consider a solid region of Euclidean 3-space, or more precisely, a compact, connected 3-dimensional submanifold $U \subset E^3$ bounded by a smooth oriented surface $\Sigma = \partial U$. Very ...
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Understanding why the Adams Spectral Sequence works

I am trying to learn about the Adams Spectral Sequence and my question is basically summed up in the title. More precisely, let $X$, $Y$, and $E$ be spectra. We have a homomorphism $[X,Y] \to Hom_{E^...