# 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 a homotopy group can be obtained from the equivalence classes. The simplest homotopy group is the fundamental group. Homotopy groups are important invariants in algebraic topology.

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### Examples of CW-complexes wich are 1-acyclic but no simply conected.

Hurewicz theorem states that if $X$ is a simply-connected CW-complex then $X$ is $(n-1)$-connected if and only if it is $(n-1)$-acyclic and that in this case $\pi_n(X)=H_n(X)$. Moreover, it is also ...
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### Uniqueness of comultiplication for sufficiently connected spaces of restricted dimensions

I'm working through an introductory homotopy theory book (Arkowitz) for self-study, and I'm a bit stuck on the following exercise. Not a lot of machinery is available at this point in the book. I'm ...
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### Proof of characterization of $E_1$ page of the Adams spectral sequence

I am following the nLab notes on the Adams spectral sequence, as they seem to be the most detailed I can find. That being said, I am still struggling to understand many of the steps. I have explained ...
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I am struggling to understand part of the top answer here: Prove homotopic attaching maps give homotopy equivalent spaces by attaching a cell A space $A$ is glued onto a general topological space $X$ ...
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### Sufficient conditions for retractions along simplices to result in "nice" complex

Suppose we have a connected simplicial complex $X$ such that we can partition the vertex set of $X$ into disjoint $M_1,\ldots,M_k$ such that for all $i$, the induced simplicial complex on the vertices ...
1 vote
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### The path components of $Map(X,Y)$ is one to one corresponding to $[X,Y]$, the set of homotopy classes between $X$ and $Y$.

Show that The path components of $Map(X,Y)$, equipped with compact-open topology with a subbasis $$\mathcal{O}_{K,U}:=\{f\in Map(X,Y): f(K)\subseteq U\},$$ is one to one corresponding to $[X,Y]$, the ...
1 vote
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### Can any co-H space be represented as a suspension? [closed]

Given a co-H space $Y$, does there exist a space $X$ such that $Y\simeq \Sigma X$? Dually, given a H space $Z$, does there exist a space $W$ such that $Z\simeq \Omega W$? If not, any other relations ...
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### Sequential spectra formed from applications of B

I am interested in the Ω-spectrum Xᵢ, Xᵢ ≅ $ΩX_{i+1}$, where $Xᵢ := BⁿX$ for a CW-complex $X$. This construction is left adjoint, right? It seemed like it wouldn't be very hard to construct the smash ...
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### Nerve Theorems for Open Coverings [closed]

There are numerous Nerve theorems that come down to something like this: For an open cover U of a space X, let N(U) be the nerve. Then under certain conditions, N(U) is homotopy equivalent to X. The ...
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### Representation of topological K-theory via Brown representability

We know that topological K-theory is a generalized cohomology theory, and reduced K-theory is a reduced cohomology theory. Thus, both are representable with a sequence of pointed homotopy functors, ...
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### Calculating a specific example of a simplicial resolution of an algebra

In the Stacks Project Tag 09D4, there's an explicit description of a simplicial resolution \begin{align*} P_{2}\to P_{1}\to P_{0} \end{align*} of an $A$-algebra $B$. I am trying to follow this ...
1 vote
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### Is an epimorphism fibered in contractible kan complexes an acyclic kan fibration?

Suppose $f : X \to Y$ is a morphism of simplicial sets which is degreewise surjective and where the fiber over any vertex of $Y$ is an acyclic kan complex. Is $f$ necessarily an acyclic kan fibration? ...
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### Spaces with two non-trivial homotopy groups

I'm wondering if there is any elementary example of a space with precisely two non-trivial homotopy groups. Let $X$ be a connected CW complex with precisely two non-trivial homotopy groups $\pi_p$ and ...
1 vote
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### Slick proof that the inverse homotopy continuous?

While trying to learn algebraic topology from Hatcher, I frequently come across maps which are "obviously continuous" but for which it would seem tedious to actually show continuity by ...
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### Understanding the proof of $4I.1$ in Hatcher's

The statement. If $X,Y$ are $CW$ complexes, then $\Sigma (X\times Y)$ is homotopically equivalent to $\Sigma X\vee \Sigma Y\vee \Sigma (X\wedge Y)$. Here $\Sigma$ means the reduced suspension. The ...
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### Definition of Homotopy [duplicate]

If $f$ and $g$ are continuous maps from $X$ to $Y$, then a homotopy $H$ between $f$ and $g$ is a continuous map from $X \times [0,1]$ to $Y$ such that $F(x,0)=f(x)$ and $F(x,1)=g(x)$ for each $x \in X$...
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### A star-shaped region is homotopic equivalent to a single point set

I am having trouble solving a problem that is probably quite simple. I want to show that an open star-shaped convex set is equivalent to a single point set $\{x_0\}$, with $x_0$ the one in the ...
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### A map between suspensions which is a homology isomorphism

Problem: $Y \subset X$ be a subcomplex of a C. W. complex. Suppose that there is a retraction $r: X \to Y$. Prove that there is a map from $\Sigma X \to \Sigma Y \vee \Sigma(X/Y)$ which is a homology ...
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### Three questions about Wikipedia's definition of Van Kampen's theorem for fundamental groups

I'm studying Algebraic Topology off of Hatcher and (unfortunately as usual) I find his definition and explanation of Van Kampen's theorem to be carelessly written and hard to follow. I happen to know ...
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### Why are the sheets the path connected components?

In Bredon's Topology and Geometry we find the following definition for covering spaces and maps: A map $p: X \to Y$ is called a covering map (and $X$ is called a covering space of $Y$) if $X$ and $Y$ ...
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### Minimal cell structures of relative CW complexes

I learned from Hatcher's book Algebraic Topology Section 4.C that we can know the mininal cell structure of a simply connected CW complex from its homology groups. The theorem is as follows: ...
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I am studying the chapter on Degree theory and homotopy invariance theorem in the context of solving non-linear systems of equations from Iterative Solution of Nonlinear Equations in Several ...
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### The fundamental group of the loop space of $(X,x_0)$ with the base point chosen not to be the constant loop in $x_0$

My question is pretty simple although I have not been able to find an answer yet: Let $c_{x_0}\in\Omega(X,x_0)$ denote the constant loop in $x_0$. Then, by standard homotopy theoretical arguments, we ...
1 vote
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### Application of Whitehead's theorem to show homotopy equivalence

Let $X$ be a CW complex satisfying $\pi_0(X) = \pi_1(X) = 0$, $H_2(X) \cong \mathbb Z^2$, and $H_j(X) = 0$ for $j \ne 2$. I am trying to prove that $X$ is homotopic equivalent to $S^2 \vee S^2$. By ...
1 vote
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### On the algebraic classification of homotopy 1-type

I'm working on Eilenberg-Mac Lane spaces and I'm reading this article in ncatlab. First, in Hatcher, a homotopy type is simple a homotopy equivalence, i.e, two spaces $X$ and $Y$ has the same ...
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### Degree of a map which is not a path

We learned in our class about the map $deg: \pi_1(S^1,1) \to \mathbb{Z}$. Which works as follows: take $\alpha$ lookat $1$ in $S^1$ take the unique lifting beginning at $0$ with respect to the ...
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### When is a freely contractible space also based contractible?

There are spaces, for example the cone of the Hawaiian earring, which are contractible but that have a basepoint such that no contraction fixes that basepoint. Are there any good sufficient conditions ...