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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What is known about homotopy groups of spheres?
I'm looking for a list/table/survey of what is known (and what is not known) about homotopy groups of spheres, for example: which are known, which are known stably, which are known primally, non-$0$ ...
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Kan fibrant replacement for a sphere
To compute the simplicial homotopy group of a space $X$, we find a Kan fibrant replacement $X \to Y$ and calculate for that for $Y$, which can be implemented in a computer program.
Computing homotopy ...
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Open conjectures and expected applications of homotopy theory to arithmetics
I hope this question is not too broad to be asked here; if it is, please feel free to close the question.
I'm currently near the end of my masters studies and subsequently search for a particular ...
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A stronger generalized Jordan curve theorem
The generalized Jordan curve theorem is usually stated as such:
Given $X\subseteq S^n$ such that $X$ is homeomorphic to $S^k$, $$\tilde{H}_i(S^n\setminus X)\cong\begin{cases}\mathbb{Z},\quad i=n-k-...
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commutative "subalgebras" of associative ring spectra
A bit of context: for any ordinary associative algebra $A$ and element $x \in A$, the subalgebra spanned by the powers of $x$ is commutative. In the universal example, this says that the free ...
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Difference between the diffeomorphism classification of a manifold $M$ and the set of equivalences of homotopy smoothings $hS(M)$
In Lopez de Medrano "Involutions on manifolds", a homotopy smoothing of a Poincaré space $X$ is a homotopy equivalence $f:M^n\rightarrow X$, where $M^n$ is a smooth $n$-dim. manifold (everything is ...
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Is this a stack?
A continuous map $f:X\to Y$ and a vector bundle $E\to X$ seem to give rise to a presheaf of groupoids on $Y$ along the following lines. For an open $U\subseteq Y$, each section of $f$ over $U$ (i. e. ...
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Topological Hochschild homology using equivariant orthogonal spectra
In the Hesselholt-Madsen paper "On the K-theory of finite algebras over Witt vectors of perfect fields", the authors develop some results concerning the Topological Hochschild homology (THH) of ...
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Detecting weak equivalence on free loop space homology
Given $f:X \to Y$ a continuous map between two spaces (unpointed CW-complexes) such that $f$ induces an isomorphism in homology with integer coefficient, and $f$ induces an isomorphism on homology of ...
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Linearity of a dg category $C$ over $HH^0(C)$
Let $C$ be a pre-triangulated dg-category over a field $k$ whose Hochschild cohomology groups $\operatorname{HH}^*(C)$ are concentrated in non-negative degree (cohomologically). Is $C$ Morita ...
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Every homotopy class contains at least a harmonic representative
Let $(M^3,g)$ be a closed, connected and oriented Riemannian $3$-manifold. A circle-valued map $v : M \to S^1$ is harmonic iff the gradient $1$-form $\omega_v = v^* d\theta \in \Omega_1(M)$ is ...
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Sullivan minimal model in the case of $H^1(V)\neq 0$
Is there a simple construction of a Sullivan minimal model $\Lambda U \rightarrow V$ in the case that $H^1(V)\neq 0$? Do you have a reference? I envisage a degree-wise construction as in the case of $...
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Bousfield-Kan and Generalized Eilenberg-Moore spectral sequences
Building on the work of Anderson and Rector, Bousfield's paper "On the homology spectral sequence of a cosimplicial space" constructs a spectral sequence which takes in a cosimplicial space (here ...
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$\ell$-adic schematic homotopy type
In the groundbreaking paper Champs Affines (DOI), Toen constructs a generalisation of rational homotopy types which he calls schematic homotopy types. This is part of a larger
programme of a theory ...
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Chains and homotopy type
Let $$C^{\ast}:\mathbf{sSet}\rightarrow E_{\infty}\text-\mathbf{dgAlg}$$
be the cochain contravariant functor from the category of simplicial sets to the category of $E_{\infty}$-dg-algebras (over $\...
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What is the definition of homotopy flat connections?
What is a definition of a homotopy flat connection - in the context of differential forms with values in a dg algebra
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Thom spectrum in the definition of power operations
I am reading now Tyler Lawson's $E_n$ ring spectra and Dyer-Lashof operations form the Handbook of Homotopy Theory and I've got a question on the Remark 1.4.19.
We have an operad $\mathcal{O}$ and $\...
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What is the fibrant replacement of an Eilenberg-MacLane space in unstable motivic homotopy theory?
One can take an Eilenberg-MacLane space $K(A, n)$ for some abelian group $A$, and view it as (locally) constant simplicial sheaf on $Sch/k$, the category of schemes, or smooth schemes, over a field $k$...
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Split cofibrations up to quasi-isomorphism
$R$ a ring $(1\neq 0)$, $\mathbf{Perf}(R)$ is the category of perfect complexes (of right $R$-modules).
Suppose that $A_{\bullet}\rightarrow B_{\bullet}\rightarrow B_{\bullet}/A_{\bullet}$ a short ...
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Infinity local systems
I have seen many references in the (geometric representation theory, symplectic geometry, etc) literature to "infinity local systems".
From what I've been told, given a good cover $\{U_i\}$ of $X$, ...
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Abelian versions of straightening and unstraightening functors
Let $X$ be a quasi-category (an inner Kan complex), let $\mathfrak{C}(X)$ be its rigidification (its associated simplicial category). J. Lurie in "Higher Topos Theory" proved the following theorem 2.1....
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Contractible chain complex from non-contractible space
Recall that a chain complex $(C_*,d)$ of abelian groups is contractible if it is homotopic to the zero map. Or equivalently: there exists a degree 1 map $F: C_* \to C_*$ such that $\operatorname{Id}= ...
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Map which is null-homotopic on compacts
This is the missing ingredient towards answering my previous question.
Let $M$ and $N$ be path connected locally compact, locally contractible metric spaces (you may assume that they are manifolds). ...
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What is the precise relationship between pyknoticity and cohesiveness?
Pyknotic and condensed sets have been introduced recently as a convenient framework for working with topological rings/algebras/groups/modules/etc. Recently there has been much (justified) excitement ...
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Monomorphisms in homotopy categories
In hTop, insertions into coproducts $i : X \to X \amalg Y$ are monomorphisms, since, for any $f,g : Z \to X$ such that $if \simeq ig$, the homotopy factors through $i$ and thus $f \simeq g$.
This ...
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A sectionwise fiber sequence is homotopy fiber sequence?
Let $\mathscr{C}$ be a site and $\mathsf{sPre}(\mathscr{C})$ the category of simplicial presheaves on $\mathscr{C}$ equipped with Jardine's local model structure. Let $E\to B$ is a sectionwise Kan ...
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Spherical objects and K-theory
My question goes as follows: given a ring $R$ (with $1\neq 0$). Define $\mathbf{Perf}_{R}$ the the category of Prefect complexes over $R$. I want to prove that the Waldhausen $K$-theory of the ...
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Conceptual definition of derived functors allowing for quick proof of comparison theorems for sheaf cohomology
There are several approaches of increasing sophistication and simplicity to defining derived functors. I know of universal $\delta$-functors and Kan extensions along localizations. More definitions ...
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Pro-trivial cosimplicial tower of spaces
Let $\{X^\bullet_s\}$ be a cosimplicial tower of spaces. In other words, for each fixed "tower degree" n, we have a (let's assume Reedy fibrant) cosimplicial space $X^\bullet_n$, and for each fixed ...
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Geometric interpretation of nonconnective, non-coconnective chain complexes / spectra?
Let's stipulate that
Connective -- i.e. nonnegatively-(homologically)-graded -- chain complexes have a very natural geometric interpretation: by the Dold-Kan theorem, they are a way of thinking about ...
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Where can I read about non-principal obstruction theory?
Most treatments of obstruction theory assume a principal Postnikov tower. I have the vague sense that if one uses cohomology with local coefficients, one does not need to make any assumptions on one's ...
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Conditions for certain inclusion functor to preserve internal homs
Suppose that $\mathcal{C}$ is a locally cartesian closed right proper Quillen model category for which all objects are fibrant. Let $x$ be an object of $\mathcal{C}$. Let $\mathbb{F}$ denote the class ...
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Describing the THH of function spectra?
Are there any results describing the $THH$ of spectra of the form $F(X, E)$ where $X$ is a space (say, finite CW) and $E$ is a (nice enough) ring spectrum?
I'm happy to put various (further, or ...
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Rational homotopy theory [closed]
I am trying to read the paper "Rational homotopy theory " by Quillen and am stuck with the notion of complete augmented algebra. He had defined the complete augmented algebra and I don't understand ...
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About the cohomology of $BG^\delta$. Making a Lie group discrete
Let $G$ be a connected Lie group. Recall that the topological group $G^\delta$ is $G$ endowed with the discrete topology. The inclusion $G^\delta \to G$ induces a map between the classifying spaces $\...
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Descent properties for rational topological cyclic homology
Descent properties can be extremely useful for studying $\operatorname{TC}$ (topological cyclic homology), since it is a sheaf in many well behaved topologies.
I was wondering what is known about $\...
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$\mathbb Z \otimes_\mathbb S \mathbb Z$ is concentrated in degree $0$ : mistake in the argument
I'm not sure this is research level so if this is not appropriate, feel free to move the question to StackExchange. However, I post it here since my "fake proof" is based on a (recent) paper and I'm ...
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What is a morphism of ∞-sites?
Recall that a morphism of sites
is a covering-flat functor
that preserves covering families.
Morphisms of sites can be identified with those
geometric morphisms of induced toposes
for which the ...
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Complex cobordism and Chern numbers
Let $L$ be the Lazard's universal ring, and $R=\mathbb{Z}[b_1,b_2,\cdots,b_n,\cdots]$, regarded as a graded ring with the degree of $b_i$ equal to $2i$. Let $\theta: L\rightarrow R$ be the ...
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$G$ uncountable implies $K(G,1)$ is not a finite CW complex
I have read that $H^i(K(\mathbb{R},1)$) has rank $2^\omega$ for any $i\in \mathbb{N}$ (see Thurston's comment here Nontrivial finite group with trivial group homologies?) therefore $K(\mathbb{R},1)$ ...
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Homological and homotopical equivalence of complex analytic varieties
Consider a map between two complex analytic varieties of finite type $f:X\to Y$. Suppose that $f$ induces isomorphisms on cohomology with (constant) integral coefficients. Under what reasonable ...
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Direct image and infinite suspension
I have a basic doubt regarding infinite suspension functor and the direct image. I write it for schemes but I guess it works the same for the topological setting so I welcome answers also from the ...
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Representing simplicial homotopy classes by empty cubes
I am looking for references concerning the following facts, which I believe to be true:
In a Kan complex $K$, can I use "empty cubes" $\partial(\Delta^1)^{(n+1)}$ with the vertex $(0,0,\ldots,0)$ ...
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Bordism groups of $X$, Thom isomorphism and characteristic numbers
Recap: bordism group
An oriented singular $n$-manifold in $X$is a map $f:M^n\to X$ where $M$ is a finite disjoint union of $n$-dimensional smooth manifolds.
The empty set is an admissible oriented ...
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The interaction between differentials on a graded ring and chain-homotopy equivalences
I am wondering about the following question:
Given a differential graded algebra $A$, how many other differentials can we put on the underlying graded ring of $A$, which are also chain-homotopy ...
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Understanding the adjunction $\mathfrak{C}:\mathbf{Set}^{\Delta^{op}}\rightleftharpoons \mathbf{Cat}_\Delta:\mathcal{N}$
Let $\mathbf{Cat}_\Delta$ denote the category of simplicially enriched categories (ie simplicial objects in $\mathbf{Cat}$ with all face and degeneracy maps bijective on objects). There is a canonical ...
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Trivial homology with local system
Let $X$ be the classifying space of the Higman group $G$. It is well known that $G$ is an acyclic group
$$H_{\ast}(X;\mathbb{Z})=H_{\ast}(pt;\mathbb{Z}).$$
Now, suppose that $\mathcal{M}$ is a ...
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Are simplicial abelian sheaves fibrant?
I want to show that simplicial abelian sheaves are fibrant. For this, I wonder whether a morphism between simplicial sheaves is a fibration iff it has RLP w.r.t. all morphisms like
$$\Lambda^n_k\times ...
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Generalized "Homology Whitehead" -- How much does stabilization remember?
Classically, the (non-local-coefficients) homology Whitehead theorem says that if $X \xrightarrow f Y$ is a map of simple spaces, and if the induced map $H_\ast(X;\mathbb Z) \to H_\ast(Y;\mathbb Z)$ ...
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Finiteness results in the category of schemes up to $\mathbb{A}^1$-homotopy
In algebraic geometry, we know that there exist geometrical conditions on a scheme $X/k$ for having finitely many rational points when $k$ is a number field. Namely for curves there is the Mordell ...