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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Model Structure on Constant-free Symmetric Operads
I am currently trying to find a reference for the assertion that the category of positive / constant-free (meaning $\cal{O}(0)=\emptyset$ is the initial object) symmetric operads $\operatorname{Opd}_\...
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Contractibility of the based path space.
Let $X$ be a topological space and $x_0 \in X.$ Consider the based path space $$\mathcal P_{x_0} (X) : = \left \{\gamma : [0,1] \xrightarrow{\text {continuous}} X\ \bigg |\ \gamma (0) = x_0 \right \}$$...
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Spectral sequence for a truncated semi cosimplicial space
Consider the simplicial indexing category $\Delta$. Now, let's denote the subcategory consisting of injections as $\Delta_{inj}$. When we're dealing with a cosimplicial space, which is essentially a ...
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Bilinear pairing on homotopy groups
Let $X,Y,Z$ be pointed spaces, and $f:X \wedge Y \rightarrow Z$ a map.
Then, $f$ induces a bilinear pairing $\pi_n(X) \times \pi_m(Y) \rightarrow \pi_{n+m}(Z)$ ($n,m \geq 1$).
I see what the pairing ...
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The Mackey functor $\underline{\pi}_n(X)$
Let $G$ be a finite group and $X$ a pointed $G$-space.
The assignment $G/H\to \pi_n(X^H)$ should define a Mackey functor. I am trying to figure out what the transfers and restrictions are.
If $H\...
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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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Adapting a deformation retraction onto an adjunction space
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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Webeil's Intro to Homological Algebra: does theorem 10.6.3 implicitly reindex cochain complexes to chain complexes?
Let $R$ be a ring, let $\mathbf{D^-(R-mod)}$ denote the derived category of bounded above cochain complexes of $R$-modules, and consider the total tensor product functor
$$
\otimes_R^\mathbf{L}: \...
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Pushout of equivalences along cofibrations are equivalences
I would like to show that if $i:A\to X$ is a cofibration and $f:A\to B$ is a homotopy equivalence, then the induced map $k:X\to X\cup_AB$ is again a homotopy equivalence.
$\require{AMScd}$
$$
\begin{...
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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 ...
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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 ...
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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 ...
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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 ...
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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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Fundamental group of two spheres in a cylinder
Hi I'm a little bit stuck on this problem
Let $$S_1 =\{ (x,y,z) \in \mathbb{R}^3: x^2+(y-1)^2+z^2=1\}\\S_2 =\{ (x,y,z) \in \mathbb{R}^3: x^2+(y+1)^2+z^2=1\}$$
and $C=\{(x,y,z)\in \mathbb{R}^3: x^2+y^...
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When is a restriction of a homotopy a homotopy?
This is a rather stupid question, but I bumped into a wrong reasoning and I can't find the mistake.
Say you have a circle $\mathbb{S}^1$ in the real plane $\mathbb{R}^2$. Now, as $\mathbb{R}^2$ is ...
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Homology of non-orientable surfaces
I am rusty on the subject and trying to reconcile this statement (paraphrased from Wikipedia, so likely not precise): "If S is a non-orientable surface, then H1(S) contains a summand Z/2". ...
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Showing that the comultiplication map $E_*E\to E_*E\otimes E_*E$ is co-associative for a flat homotopy commutative ring spectrum
Let $(E,\mu,e)$ be a flat homotopy commutative ring spectrum, so we have an isomorphism
$$\Phi_E:E_*E\otimes_{\pi_*E}E_*E\to E_*(E\wedge E)$$
sending homogeneous elements $x:S^n\to E\wedge E$ and $y:S^...
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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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Geometric interpretation of pairing between bordism and cobordism
In page 448 of these notes, a pairing between bordism and cobordism
$$\langle \ ,\ \rangle: U^m(X)\otimes U_n(X)\rightarrow \Omega^U_{n-m}$$
is defined as follows. Assume $x\in U^m$ is represented by $...
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Showing that inducded maps to relative homotopy groups is 0
Let $X, Y, Z$ be based spaces and define
$F_2(X, Y, Z) = \{(x, y, z) \in X \times Y \times Z|$ at least one of $x, y, z$ is $= * \}$
Prove that the inclusion $F_2 (X, Y, Z) \to X \times Y \times Z$ ...
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Are any subalgebras of the steenrod algebra isomorphic to the group algebra over for some group? [closed]
Heading says it all. Wondering if there are any subalgebras of the steenrod algebra which are isomorphic as hopf algebras to $\mathbb{F}_2{G}$ for some group $G$? In particular interest to me are the ...
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Homotopy in path-connected space
Let $Y$ be path connected then every constant function $X\to Y$ are homotopic to each other.
I think I get what's the point of the question. I'm having some issues formally writing everything. The ...
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Stable diffeomorphisms of disks
My impression is that computing (the homotopy groups of) $\operatorname{Diff}(n)$, the diffeomorphisms of the n-disk, is a hard problem.
For example, the connected components are related to exotic ...
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Proving that an isotopy of $\mathbb{R}^{2}$ must preserve orientation of an embedded circle.
Consider two very simple maps from $\mathbb{R}^{2}$ to itself: the identity $I(x,y) = (x,y)$ and a reflection about the $y$-axis: $R_{y}(x,y) = (-x,y)$. For any embedded circle, $S^{1} = \{(x,y) \in \...
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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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If $f: \mathbb CP^2 \rightarrow \mathbb CP^2$ is a homeomorphism, then $f(\mathbb CP^1)$ intersects $\mathbb CP^1$ [closed]
Consider the standard embedding $\mathbb CP^1 \subseteq \mathbb CP^2$.
Let $f : \mathbb CP^2 \rightarrow \mathbb CP^2$ be a homeomorphism. The goal is to show that $f(\mathbb CP^1)$ always intersects $...
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A problem of non-fixed point homotopy.
Let $G$ be a topological space and $m : G \times G \longrightarrow G$ be a continuous map. Let $x_0 \in G$ be such that both the maps $m (x_0, \cdot) : G \longrightarrow G$ and $m (\cdot, x_0) : G \...
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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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Homotopy invariance seems to suggest 1 = 0 ? Please help correct my understanding.
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 ...
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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 ...
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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 ...
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Homotopy along a path
Hello I have the following question. Let $\gamma$ be a path in $X$ then we say that two loops $f_0,f_1$ are homotopic along $\gamma$ iff there is a continuos map $H:I^2 \to X$ s.t $H(t,0) = f_0(t), H(...
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About homotopies from rotation Matrizes
Consider a separable, infinite dimesnional Hilbert space $H$. I want to show that the inclusion into the odd indices $(e_1,e_2,e_3,...)\mapsto (e_1,0,e_2,0,e_3,0,...)$ is isometrically homotopic to ...
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Are Möbius transformations homotopic to the identity on the upper half-plane?
probably this question is totally easy and obvious but I am very confused at the moment. So assume we have a matrix $\gamma$ in $SL_2(\mathbb{R})$ acting on the usual upper half-plane $\mathcal{H}$ by ...
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Lemma 2.4 in chapter 2 of Goerss Jardine Simplicial homotopy theory
I am having trouble with lemma 2.4 of the chapter of simplicial categories. I have a feeling there are mistakes in the proof or maybe typos. He wants to prove the 2.1.2 property of the definition of a ...
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