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-$0$ ...
5 votes
2 answers
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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 ...
Student's user avatar
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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 ...
lush's user avatar
  • 330
2 votes
0 answers
182 views

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-...
Anonymous's user avatar
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4 votes
1 answer
313 views

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 ...
pupshaw's user avatar
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4 votes
1 answer
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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 ...
Kafka91's user avatar
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5 votes
0 answers
232 views

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. ...
მამუკა ჯიბლაძე's user avatar
5 votes
1 answer
564 views

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 ...
S.S.'s user avatar
  • 255
14 votes
1 answer
292 views

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 ...
Simon Henry's user avatar
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2 votes
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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 ...
Daniel Pomerleano's user avatar
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1 answer
371 views

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 ...
Eduardo Longa's user avatar
5 votes
1 answer
288 views

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 $...
Pavel's user avatar
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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 ...
prefix.crm114's user avatar
9 votes
1 answer
446 views

$\ell$-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 ...
Patrick Elliott's user avatar
5 votes
0 answers
408 views

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 $\...
GSM's user avatar
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6 votes
1 answer
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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
Jim Stasheff's user avatar
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7 votes
1 answer
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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 $\...
Igor Sikora's user avatar
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5 votes
0 answers
176 views

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 $k$...
user155861's user avatar
4 votes
1 answer
187 views

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 ...
Let's user avatar
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14 votes
2 answers
929 views

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 $X$, ...
user142700's user avatar
2 votes
1 answer
275 views

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....
David C's user avatar
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4 votes
1 answer
400 views

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}= ...
user155668's user avatar
6 votes
1 answer
450 views

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). ...
erz's user avatar
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29 votes
3 answers
4k views

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 ...
Emily's user avatar
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2 votes
0 answers
105 views

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 ...
monohotop's user avatar
2 votes
0 answers
91 views

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 ...
Lao-tzu's user avatar
  • 1,856
3 votes
1 answer
176 views

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 ...
Let's user avatar
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3 votes
0 answers
168 views

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 ...
Arrow's user avatar
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2 votes
0 answers
78 views

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 ...
guest's user avatar
  • 21
5 votes
0 answers
519 views

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 ...
Tim Campion's user avatar
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4 votes
0 answers
274 views

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 ...
Tim Campion's user avatar
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2 votes
1 answer
117 views

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 ...
CuriousKid7's user avatar
4 votes
0 answers
151 views

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 ...
John Greenwood's user avatar
2 votes
0 answers
290 views

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 ...
GURI920826's user avatar
9 votes
1 answer
525 views

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 $\...
Overflowian's user avatar
  • 2,523
5 votes
0 answers
190 views

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 $\...
Noah Riggenbach's user avatar
5 votes
0 answers
231 views

$\mathbb Z \otimes_\mathbb S \mathbb Z$ 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 ...
Maxime Ramzi's user avatar
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8 votes
0 answers
335 views

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 ...
Dmitri Pavlov's user avatar
11 votes
1 answer
477 views

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 ...
Xing Gu's user avatar
  • 935
3 votes
1 answer
316 views

$G$ 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 $K(\mathbb{R},1)$ ...
Overflowian's user avatar
  • 2,523
7 votes
0 answers
219 views

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 ...
W. Rether's user avatar
  • 395
4 votes
0 answers
190 views

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 ...
Tintin's user avatar
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3 votes
1 answer
87 views

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 $(0,0,\ldots,0)$ ...
Daniel Robert-Nicoud's user avatar
3 votes
1 answer
538 views

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 ...
Overflowian's user avatar
  • 2,523
5 votes
0 answers
123 views

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 ...
Mo Behzad Kang's user avatar
6 votes
1 answer
421 views

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 ...
user avatar
5 votes
1 answer
649 views

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 ...
lun's user avatar
  • 69
4 votes
1 answer
191 views

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 ...
Nanjun Yang's user avatar
10 votes
1 answer
444 views

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 $H_\ast(X;\mathbb Z) \to H_\ast(Y;\mathbb Z)$ ...
Tim Campion's user avatar
  • 60.6k
3 votes
0 answers
178 views

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 ...
curious math guy's user avatar

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