Questions tagged [homotopy-theory]

Homotopy theory is an important sub-field of algebraic topology. It is mainly concerned with the properties and structures of spaces which are invariant under homotopy. Chief among these are the homotopy groups of spaces, specifically those of spheres. Homotopy theory includes a broad set of ideas and techniques, such as cohomology theories, spectra and stable homotopy theory, model categories, spectral sequences, and classifying spaces.

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Notation A(g,n) related to Eilenberg MacLane spaces [on hold]

What does A(g,n) mean in this context (as opposed to K(g,n))? (In context of this video: https://www.youtube.com/watch?v=STSDjtY4cm4)
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Descent properties of topological Hochschild homology

Question: What is the finest topology on $\mathrm{CAlg}$ (commutative ring spectra) for which THH (Topological Hochschild Homology) satisfies descent? Adaptations of the arguments appearing in ...
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Definition of hypercover for simplicial presheaves and hypercovering in $\infty$-topos

By lemma 4.9 in Dugger-Hollander-Isaksen, a hypercover is defined as an augmented simplicial object $U_\bullet\to X$ in the category of simplicial presheaves such that each $U_n$ is a coproduct of ...
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Homotopy fibre sequence and left Bousfield localization

Let $\mathcal{M}$ be a pointed model category and $\mathcal{C}$ a class of maps in $\mathcal{M}$ for which the left Bousfield localization ${\rm L}_{\mathcal{C}}\mathcal{M}$ exists (see Hirschhorn, ...
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Is the inclusion $\Delta^{op} \to \Gamma^{op} = Fin_\ast$ homotopy cofinal?

There's a canonical functor $i: \Delta^{op} \to Fin_\ast$. For example, one uses the pullback $i^\ast$ to turn a $\Gamma$-space into a simplicial space, and then takes a geometric realization to ...
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Descent in the injective model structure and descent for simplicial presheaves

In Jardine's book Local Homotopy Theory P.102, a simplicial presheaf $X$ on a site $C$ is said to satisfy descent if some local injective fibrant replacement $ j : X → Z $ is a sectionwise weak ...
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Derived base change in étale cohomology

Given a commutative square of ringed topoi $$\begin{array}{ccc}X'\!\! & \overset{f'}\to & Y'\!\! \\ \!\!\!\!\!{\small g'}\downarrow & & \downarrow{\small g}\!\!\!\! \\ X & \...
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Does a homotopy sheaf functor commute with group completion

Let $\text{sPre}(C)$ be the category of simplicial presheaves with Grothendieck topology $\tau$. Let $\pi_n^{\tau}$ be the $\tau$-homotopy sheaves defined as follow Does $\pi_n^{\tau}$ commute with ...
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Koszulness of some DG-algebras and a paper by Kohno and Oda

This is a follow-up of my previous question Formality of the 2nd ordered configuration space of a closed Riemann surface. At page 131 of [B], R. Bezrukavnikov states Proposition 4.1, in which he ...
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The homotopy theory presented by a Waldhausen category

Waldhausen introduced his categories for the purposes of defining algebraic $K$-theory of suitable categories. From a modern perspective, it looks like he was really doing two things at once: ...
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Which spaces are most naturally presented simplicially?

It's often a great technical convenience to know that you can "do homotopy theory" with simplicial sets. But if you really get down to it geometrically, it can be awkward. In general, if I have a CW ...
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$K_3(\mathbb{Z})$ and $\pi ^S_3$

This is an afterthought on this MO question, and also on Gannon's book mentioned there, about $K_3(\mathbb{Z})=\mathbb{Z}/48$. Neither the question nor the book mentions a possible connection with the ...
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A particular pushout of homologicaly rational spaces

Let $R^{\delta}$ be the topological group of additive real numbers (with discrete) topology and let $R$ be the topological group of additive real number with the standard topology. Let $X$ be a (...
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Open subspaces of CW complexes

I am looking at the paper Covering homotopy properties of maps between CW complexes or ANRs by Mark Steinberger and James West and a claim is made in the proof of their first main theorem ...
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Proof that $Sing^IX$ is $I$-invariant for an interval object in a site by “simplicial decomposition”

I just saw a proof showing that $Sing^{A^1}X$ is $A^1$ invariant where they use an algebraic topological analogue of ``simplcial decomposition'' defined as follow: The argument works by showing that ...
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Is transfinite composition of weak equivalences of simplicial presheaves a weak equivalence?

In a left Bousfield localization of the projective model structure on the category of simplicial presheaves, what is the condition that transfinite composition preserves weak equivalences? How about ...
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Homology of a loop-suspension space and action of $\mathcal{D}_1$-operad

If $X$ is a based connected topological space, it is well-known what the homology of $\Omega\Sigma X$ is: according to the Bott-Samelson theorem, it is a tensor algebra over reduced homology of $X$. (...
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Which homotopy types can be realized as the classifying space of a right-cancellative discrete monoid?

McDuff showed that every connected homotopy type can be realized as the classifying space of a discrete monoid, but the monoid she constructs has lots of idempotents. Question: Which homotopy types ...
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Are simplicial finite CW complexes and simplicial finite simplicial sets equivalent?

Edit Originally the question was whether an arbitrary diagram of finite CW complexes can be approximated by a diagram of finite simplicial sets. In view of Tyler's comment, this was clearly asking for ...
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Group completion of $E_k$-algebras

Let $X$ be an $E_k$-algebra. We can form the delooping $BX$, which is a $E_{k-1}$-algebra. The space $\Omega B X$ is again an $E_k$-algebra, which is grouplike (i. e. $\pi_0(\Omega B X)=\pi_1(B X)$ is ...
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Are the real and complex Adams operations compatible under the inclusions $U(n) \rightarrow SO(2n)$?

Does the following diagram commute? $$ \require{AMScd} \begin{CD} BU @>{\psi^k}>> BU \\ @VVV @VVV \\ BO @>{\psi^k}>> BO \end{CD} $$ Evidence for: $rc = 2$, it works for $BU(1) \...
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Weak homotopy equivalence of sites

There are several notions of weak homotopy equivalence for topological spaces. The standard one can be formulated as follows: a map of spaces $X\to Y$ is a homotopy equivalence if the map of ...
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Cyclic homotopies of quotients of $S^3$

We are given a free action of an abelian finite group on $S^3$. Let $L$ denote the quotient space and let an element $\alpha \in \pi_1 L =G$ be given. Does there exist a cyclic homotopy $h_t:L \to L$ ...
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Homotopy pullback of $\mathbb{A}^1$-projections in the Nisnevich localization

Let $L_{Nis}(sPre(Sm_S))$ be the Nisnevich localization of the category of simplicial presheaves, how to see that whether $\mathbb{A}^1$-projections $\mathbb{A}^1\times_S X\to X$ are closed under ...
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In a subset of $\mathbb{R}^2$ which is not simply connected does there exist a simple loop that does not contract to a point?

I previously asked In which topological spaces does the existence of a loop not contractable to a point imply there is a non-contractable simple loop also? Given the broad scope of this question I ...
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In which topological spaces does the existence of a loop not contractible to a point imply there is a non-contractible simple loop also?

In another MathOverflow post I asked: In a topological space if there exists a loop that cannot be contracted to a point does there exist a simple loop that cannot be contracted also? Note that ...
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In a topological space if there exists a loop that cannot be contracted to a point does there exist a simple loop that cannot be contracted also?

I'm interested in whether one only needs to consider simple loops when proving results about simply connected spaces. If it is true that: In a Topological Space, if there exists a loop that cannot ...
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Why does every chain complex have a map into its cone?

In Weibel's An introduction to homological algebra he defines a cone as an explicit chain complex associated to the given one -i.e. for a chain $C=(C_i, d)$ he defines $Cone(C)=\left(C_{i-1} \oplus ...
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Fibrant objects in Bousfield localization of homotopy pullback closure of Nisnevich hypercovers

Let $M$ be a model topos and $S$ a set of morphisms, there exists a set of morphism $\bar{S}$ which is generated by the $S$-local equivalences which is closed under homotopy pullbacks in $M$. Suppose ...
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Stable splitting of products

This question concerns the well-known homotopy equivalence $$ \Sigma (X\times Y) \simeq \Sigma (X \vee \ Y) \vee \Sigma (X\wedge Y) $$ (I'm happy to use only CW complexes). I can see that there is ...
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Equivalent definitions of Thom spectra

Background and notations: Recall the classical contruction and definition of Thom spectra. To a spherical fibration $S^{n-1} \to \xi \to B$, we can associate the data of a Thom space $T_n(\xi)$, given ...
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$L_\infty$-quasi inverse for the contravariant Cartan model on principal bundles

First of all I want to apologize for the much too long post. A Lie group $G$ is acting on a smooth manifold $M$, then we define \begin{align*} T^k_G(M)= (S^\bullet \mathfrak{g}\otimes T^k_\mathrm{...
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Diffeomorphism type of the added sphere in simply connected surgery

A classical result of simply connected surgery theory, is that if two normal maps $f:M_i\rightarrow X$, $i=0,1$ are normally cobordant and if the dimension of the manifolds is odd, there exists a ...
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Homotopy equivalence of $K$-theory and $G$-theory

Let $X$ be regular variety then it is known that $Q(Vect(X))\cong Q(\mathcal{M}_X)$. Where $Q$ is the Quillen's q-construction and $\mathcal{M}_X$ is the category of coherent sheaves on $X$. You can ...
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What is the intuition for higher homotopy groups not vanishing?

The homotopy groups of the spheres $S^n$ (see Wikipedia) vanish for the circle $S^1$ as, naively speaking, there are not higher order holes to be grasped by higher order homotopy groups. This ...
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Pushforward of an internal category along a functor

Let $F:C\to D$ be a “nice” functor (for example, $H_*(-;\mathbb{Z}):\mathbf{Top}\to \mathbf{Ab}^{\mathbb{Z}}$). Now assume that we have a category $O$ internal to $C$. Is there a canonical way to ...
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Replacing the Fibre of a Fibration

This was a question I first asked on stack exchange, here. In my head it seems like a fairly reasonable thing to ask for, but I'm not aware of any construction in the literature. Let $p:E\rightarrow ...
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Is there a weak homotopy equivalence between Sp(2n,ℂ)/U(n) and SU(n)?

This question, Is there a weak homotopy equivalence between Sp(2n,ℂ)/U(n) and SU(n)?, is at the end of a long string of my comments in https://math.stackexchange.com/questions/3296373/is-sp2n-mathbbc-...
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Crystals and nilpotence

Fix a prime $p$ and a height $n \geq 1$, then there is a closed substack $\mathcal{M}^{\geq n}$ of the stack of $\mathbb{F}_p$-formal groups consisting of formal groups having height $\geq n$. A ...
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Categorical Significance of Fibrations

It is well known that the category $\text{Set}$ classifies covering spaces among $1$-categories. That is, for each topological space $X$, there is an equivalence of categories $[ \Pi (X) , \text{Set}]...
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Topological invariants of a certain “stratified” manifold, with pieces of different “dimensions”

Disclaimer: I don't fully understand what I'm talking about in the question below. I'm still trying to figure out the right question to ask. Quotations and question marks in brackets mean that I'm not ...
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Homotopy pullbacks and pushouts in stable model categories

There are lots of similar questions that have been answered on this topic (particularly Homotopy limit-colimit diagrams in stable model categories), but I have a specific question that I do not ...
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Cyclic version of Lie algebra cohomology

Lie algebra cochains have a differential $d$ where $d^2 =0$ because of the Jacobi identity, which can be written in the cyclic form or the Leibniz form. $L_\infty$ algebra cochains have a ...
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Obstruction to homotopy, cohomology operations and Dold-Whitney theorem

I am reading the famous paper by Dold and Whitney "Classification of Oriented Sphere Bundles Over A 4-Complex". I'll state their theorem for the case of SO(3) bundles Classification Theorem:Let $B_1,...
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Cardinalities associated to the Bousfield lattice

By Ohkawa's theorem, the Bousfield lattice $B$ (of the $\infty$-category of spectra) is a small, complete lattice with $2^{\aleph_0} \leq |B| \leq 2^{2^{\aleph_0}}$ (the exact cardinality is an open ...
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Other homotopy invariants?

The idea of using maps from a sequence of simple standard objects into a topological space $X$ as $probes$ to explore its topology is ubiquitous. One organizes these maps into equivalence classes in ...
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Homotopy colimits of long sequences

Let $\lambda$ be a limit ordinal, and let $F: \lambda\to \mathcal{T}_*$ be a diagram of pointed spaces with shape $\lambda$. Write $X = F(0)$ and $Y = \mathrm{hocolim} F$. I believe it to be true (I ...
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Is it always possible to write a derived manifold (in the sense of Spivak) as a homotopy colimit of principal derived manifolds?

Is it always possible to write a derived manifold as a homotopy colimit of principal derived manifolds (i.e. zero sets of smooth functions)? This is true for schemes and derived schemes, so it seems ...
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Dold-Kan correspondence in the category of symmetric spectra

The Dold-Kan correspondence between the category of simplicial abelian groups and the category of non-negatively graded chain complexes of abelian groups is a classical result. It states that the ...
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1answer
196 views

A model for the framed little disks operad $f{\cal D}_n$ with arity one *equal* to $SO(n)$?

The framed little disks operad $f{\cal D}_n$ can be described as the semidirect product ${\cal D}_n \rtimes SO(n)$, where ${\cal D}_n$ is the little disks operad and $SO(n)$ is the special orthogonal ...