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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$$ \left(\frac{\text{Man}^{\text{fr}}}{\text{Cobordism}},\coprod,\times \right)\simeq \left((\text{Fin}^{\simeq},\coprod)^{\text{gp}},\times\right)?$$ [closed]

If we combine a theorem of Pontryagin and the Barratt-Priddy-Quillen theorem we get that both sides of $$ \left(\frac{\mathrm{Man}^{\mathrm{fr}}}{\mathrm{Cobordism}},\coprod,\times \right)\simeq \left(...
Ola Sande's user avatar
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Does there exist a closed 4-manifold whose $\pi_2$ contains torsion elements

Does there exist a closed 4-manifold $X$ such that $\pi_2(X)$ contains torsion elements? And, if so, does there exist a closed 4-manifold $X$ such that $\pi_2(X)\neq 0$ but $\pi_2(X)\otimes \mathbb{Q}=...
Boyu Zhang's user avatar
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0 answers
39 views

How is the behaviour of a deformation retract under a fibration? [duplicate]

Let $p:E \rightarrow B$ a fibration and take $A\subset B$ a deformation retract of B. Is it true that $p^{-1}(A)$ is a deformation retract of E? By deformation retract I mean the weaker definition. I'...
Alvaro Sopeña's user avatar
2 votes
0 answers
70 views

Implementation of the nerve of a category in GAP

I am trying to compute the second cohomology group of the coset poset $$\mathcal{C}_G\mathcal{F}=\{gA\mid g\in G,A\in\mathcal{F}\}$$ for a finite group $G$ and a family $\mathcal{F}$ of subgroups of $...
Antoine's user avatar
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3 votes
0 answers
124 views

Is taking Freyd envelopes adjoint to taking stable module categories?

Let $T$ be an (idempotent-complete) triangulated category. Then the Freyd envelope $mod(T)$ is an abelian category, the universal recipient of a homological functor $T \to mod(T)$. The Freyd envelope ...
Tim Campion's user avatar
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11 votes
1 answer
349 views

Plus construction on Simplicial Sets?

I had asked this question in Math StackExchange a few days ago, but didn't get any answers. I believe its more suitable to be asked here. Write $\mathsf{sSet}$ for the category of simplicial sets and $...
wind's user avatar
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8 votes
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318 views

Is the Whitehead bracket $\pi_{p}(X)\otimes \pi_{q}(Y)\to \pi_{p+q-1}(X\vee Y)$ injective?

Let $X$ and $Y$ be finite CW-complexes and $p,q\geq 2$. The Whitehead bracket induces a homomorphism $\pi_{p}(X)\otimes \pi_{q}(Y)\to \pi_{p+q-1}(X\vee Y)$, $\alpha\otimes \beta\mapsto [\alpha,\beta]$....
J.K.T.'s user avatar
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9 votes
1 answer
661 views

Roadmap for Algebraic Geometry/Homotopy Theory/Algebraic $K$-Theory intersection

I’m afraid that this is quite a general question but I am hoping some experts can weigh in. I am a student generally interested in learning more about the intersection of algebraic geometry, algebraic ...
Song Ye's user avatar
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2 votes
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Infinite loop space as an endofunctor of compactly generated weak hausdorff topological spaces?

I am trying to see whether it is possible to define smash product of infinite loop spaces using the space $S^{\infty}$. Let C be the category of compactly generated weak Hausdorff topological spaces. ...
Ronald J. Zallman's user avatar
7 votes
1 answer
150 views

Reference request for equivalences between different models of lax limits

There are several models for lax limits of model categories/ $\infty$-categories in the literature. For example, within the realm of $\infty$-categories one can construct them using coCartesian ...
happymath's user avatar
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110 views

Hypercube of chain complexes as functor from (Δ^1 )^n to ∞-category of chain complexes

A hypercube of chain complexes consists of $\mathbb{Z}$-graded vector spaces $C_\epsilon$ for $\epsilon\in\{0,1\}^n$ and maps $D_{\epsilon,\epsilon^\prime}:C_{\epsilon}\to C_{\epsilon^\prime}$ for $\...
Faniel's user avatar
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3 votes
1 answer
151 views

Bⁿ and coherence

I understand that an internal abelian group in CW-complexes (while it is not the most extensive structure for which this can be done) produces a classifying space which itself has the structure of an ...
Ronald J. Zallman's user avatar
1 vote
0 answers
120 views

Duality in Spc with ∧ and [-,-]

I am thinking about two duality theorems for H-spaces and their actions. By H-space is meant a commutative monoid in the derived (homotopy) category of based connected CW-complexes. We can consider ...
Ronald J. Zallman's user avatar
7 votes
1 answer
411 views

Model categories as a tool to resolve size issues for localizing categories

I have a rather basic question about one motivation for introducing model categories in David White's notes, as a possible way to overcome troubles with size issues appearing when localizing a ...
user267839's user avatar
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4 votes
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Does the forgetful functor $F:\mathrm{CAlg}\to\mathrm{Alg}^{(1)}$ sending $E_\infty$-ring spectra to $E_1$-ring spectra preserve limits and colimits?

In remark 7.1.0.4. of Lurie's Higher Algebra, the sequence $(E_n^{\otimes})_{0\leq n\leq\infty}$ of $\infty$-operads induces forgetful functors for the sequence of categories $(\operatorname{Alg}^{(n)}...
Miso's user avatar
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3 votes
1 answer
135 views

Can a phantom map have finite cofiber?

Let $f : X \to Y$ be a nonzero phantom map between spectra. Can the cofiber of $f$ be a finite spectrum? Recall that a map $f$ is said to be phantom if $f \circ i = 0$ whenever $i : F \to X$ is a map ...
Tim Campion's user avatar
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2 votes
2 answers
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Is the mapping cylinder a replacement for morphism by cofibration in model categories?

Let $M$ be a model category, consider a very good cylinder object $X \coprod X \to X \times I \overset{\operatorname{pr}}{\to} X$ (here $X \times I$ is just a notation, no object $I$ is implied), that ...
Arshak Aivazian's user avatar
1 vote
0 answers
204 views

Properties of colim Ωⁿ Σⁿ X

I am thinking about the paper of Gaunce Lewis Jr. showing the incompatibility of a certain five desirable properties of spectra. This paper makes me curious about the properties of the endofunctor $Q: ...
Ronald J. Zallman's user avatar
2 votes
0 answers
140 views

Local-to-global philosophy for crossed modules

In the survey Groupoids and crossed objects in algebraic topology Ronald Brown made after Corollary 5.17 (p 30) an very interesting remark I not fully understand. He stated that this Corollary 5.17 ...
user267839's user avatar
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1 vote
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A question on $BP$ and $E_\infty$ models for ring spectrums

I am a beginner in this field. My question is (1) Is the existence of $E_\infty$ ring structure not closed under weak equivalence of ring spectra? (2) If (1) is true, what is the risk of replacing a ...
Miso's user avatar
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1 vote
1 answer
216 views

Symmetric-monoidal-associative smash product up to homotopy

I am thinking about sequential spectra. I am trying to figure out if the smash product here is symmetric monoidal associative up to homotopy. See definition 3.16 in the above. Recall that a sequential ...
Ronald J. Zallman's user avatar
7 votes
1 answer
107 views

Semi-simple algebras over operads

I believe people thought about this questions, however I couldn't find any reference. I appreciate if someone could direct me to some detailed discussions about it. The categories of associative ...
Li Guanyu's user avatar
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4 votes
0 answers
59 views

Homotopy fixed point spectral sequence for cooperation algebra

For a finite group $G$ acting on a spectrum $X$, there is a homotopy fixed point spectral sequence calculating $X^{hG}$ with signature $$E_2^{p, q} = H^p(G, \pi_q X) \implies \pi_{q-p}(X^{hG}).$$ ...
categorically_stupid's user avatar
7 votes
0 answers
109 views

The homotopy inverse on Quillen's $S^{-1}S$ construction

Suppose $S$ is a symmetric monoidal groupoid. Take Quillen and Grayson's $S^{-1}S$-construction, which is a symmetric monoidal category with objects given by pairs $(m,n)$ and maps given by ...
Georg Lehner's user avatar
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2 votes
0 answers
143 views

Connection on relative topological periodic cyclic homology

I have been looking Bhatt-Morrow-Scholze's paper: https://arxiv.org/pdf/1802.03261.pdf and came to a naive question. Let $C$ be a dg-category (with assumptions?) over $\mathbb{F}_p[[z]]$ and view this ...
Daniel Pomerleano's user avatar
2 votes
0 answers
195 views

Using the Dold-Thom Theorem to define \'etale cohomology

For reasonable spaces $X$, the Dold-Thom Theorem states that $\pi_i(SP(X)) \cong \tilde{H}_i(X)$ where $SP(X) = \bigsqcup_i \mathrm{Sym}^i(X)$. There is a purely algebro-geometric realization of this ...
Asvin's user avatar
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4 votes
0 answers
234 views

Homotopy group of maps into S^3 using its Lie group multiplication to define the group operation

The Bruschlinsky group of maps of a space X into S1 up to homotopy, using the multiplication on S1, is well-known to equal the first cohomology group of X (at least assuming X is a reasonably nice ...
Daniel Asimov's user avatar
2 votes
0 answers
135 views

A possible generalization of "homotopy" to study group actions of various kinds

This is a naive question about abstract homotopy theory by someone who knows nothing about it, except that it involves some generalization of the notion of "homotopy". If we think of $O(n)$ ...
semisimpleton's user avatar
14 votes
1 answer
837 views

What is $\pi_{23}(S^2)$?

The first $22$ homotopy groups of the $2$-sphere were worked out by Toda in 1962, but I cannot find any results extending that to any higher homotopy groups of $S^2$. Are any more of these groups ...
Joe Shipman's user avatar
6 votes
1 answer
183 views

Weakly contractible $X$, but none of the maps $*\to X$ are cofibrations

Let $\mathrm{Top}$ be the category of all topological spaces and continuous maps. The Quillen model structure on $\mathrm{Top}$ has weak equvalences $W = \{ \text{weak homotopy equivalences} \}$, ...
mathmo's user avatar
  • 161
13 votes
2 answers
939 views

Categories on which one can determine all model structures?

Famously, there are exactly nine model structures on the category of sets, which are detailed here. In this case, one can exhaustively determine all six weak factorization systems and then see which ...
Tim Campion's user avatar
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14 votes
3 answers
580 views

Strøm model structures on the category of simplicial sets

Let $X,Y$ be simplicial sets. A simplicial homotopy is a simplicial map of the form $h:X\times\Delta^1\rightarrow Y$. There are two distinguished maps $$ in_0:X\cong X\times\Delta^0\xrightarrow{1\...
Tyrone's user avatar
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5 votes
1 answer
288 views

Are there “nice” explicit representations of infinite order elements in $\pi_{4n-1}(S^{2n})$

Motivation: The representation is very clean in the Hopf fibration case (2n=2,4,8). These maps are known since Serre's 1951 paper. There are clean generators for the only other infinite case $\pi_n(S^...
David Lehavi's user avatar
  • 4,299
3 votes
1 answer
131 views

Linearity of topological periodic cyclic homology

Let $A$ be an $E_\infty$ ring spectrum, $B$ a ring spectrum. Then if I understand correctly, $TP(A)$ is a ring spectrum by the lax monoidal property of $TP$. Suppose there is a map of ringed spectra ...
onefishtwofish's user avatar
4 votes
1 answer
329 views

How to calculate $\mathrm{TP}(\mathbb{F}_p[t])$?

$\DeclareMathOperator\TP{TP}$I am trying to learn about topological periodic cyclic homology following the notes: https://www.uni-muenster.de/IVV5WS/WebHop/user/nikolaus/Papers/Lectures.pdf https://...
onefishtwofish's user avatar
6 votes
1 answer
368 views

Non-triviality of Whitehead products in wedges of CW-complexes

Suppose $X$ and $Y$ are finite, simply connected, based CW-complexes and $m,n\geq 2$. If $a\in \pi_m(X)$ and $b\in \pi_n(Y)$, then one can regard these as elements of the homotopy groups of $X\vee Y$. ...
J.K.T.'s user avatar
  • 487
15 votes
1 answer
735 views

If homotopy groups of spaces are identical, then stable ones are also identical?

Is it true that if pointed spaces $X, Y$ have the same homotopy groups $\pi_n(X) \cong \pi_n(Y)$, then they have the same stable homotopy groups $\pi^S_n(X) \cong \pi ^S_n(Y)$? In particular, is this ...
Arshak Aivazian's user avatar
2 votes
0 answers
163 views

Triviality of map $(\Sigma \theta)^*$

We know that there is a cofibration sequence $$S^{4n+1}\xrightarrow{\theta}\Sigma^{4m-1} Q_{n-m} \rightarrow \Sigma^{4m-1} Q_{n-m+1} \rightarrow S^{4n+2}\xrightarrow{\Sigma\theta}\Sigma^{4m} Q_{n-m}.$$...
Sajjad Mohammadi's user avatar
9 votes
1 answer
596 views

Homotopy groups of finite CW complex finitely generated as Lie algebra

This is probably a well-known question, but I haven't found the answer on MO or MSE. It is well-known that the homotopy groups of a finite CW complex $X$ need not be finitely presented, even as $\...
R. van Dobben de Bruyn's user avatar
4 votes
1 answer
346 views

Two spectral sequences arising from a simplicial spectrum

Let $X_\bullet$ be a simplicial spectrum, and let $X = |X_\bullet|$ be the geometric realization. Let's assume each $X_n$ is connective. From this situation, we can form two filtrations on $X$: the ...
Brian Shin's user avatar
10 votes
1 answer
408 views

Where does the definition of ($\infty$-)groupoid cardinality come from?

The cardinality of a finite set $X$ is the number of its elements. Once you know that, you would define the groupoid cardinality of a $\infty$-groupoid $X$ as the quantity $$\lvert X\rvert := \sum_{[x]...
Matthew Niemiro's user avatar
7 votes
0 answers
174 views

Complex cobordism and integrable systems

In Jack Morava's paper On the complex cobordism ring as a Fock representation, it was remarked right at the beginning that complex cobordism may play a role in the theory of integrable systems. In ...
user1271629's user avatar
4 votes
0 answers
78 views

For a map $x: S^0 \to X$, $J(X,x) \otimes Y = 0$ iff $x \otimes 1_Y$ is nilpotent?

Let $X$ be a spectrum, and let $x : S^0 \to X$ be a map. Let the stable James construction $J(X,x)$ denote the free $E_1$ ring on the $E_0$-ring $(X,x)$. It is computed as the colimit of the $\Delta^{...
Tim Campion's user avatar
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8 votes
0 answers
130 views

The James and Morse filtrations of homotopy groups

Denote by $JX$ the James construction on a path connected, well-pointed space $X$. This space is filtered by subspaces $X=J_1X\subseteq\dots\subseteq J_nX\subseteq\dots\subseteq JX$ and is the domain ...
Tyrone's user avatar
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4 votes
1 answer
231 views

On the initiality of the inclusion from the simplex category to the paracycle category

Thm B.3 of Nikolaus and Scholze shows that the natural inclusion $\Delta \to \Lambda_\infty$, from the simplex category to the paracycle category, is an initial functor, i.e. satisfies the hypotheses ...
Tim Campion's user avatar
  • 60.5k
6 votes
1 answer
411 views

Does $\pi_1(H)=0\Rightarrow \pi_3(G/H)=0$ for a simple and simply connected Lie group $G$?

$\DeclareMathOperator\SU{SU}$Let $G$ be a simple and simply-connected Lie group and $H\neq 1$ be a simple and simply connected subgroup, is it true that $\pi_3(G/H)=0$? If not, what is a counter-...
Davi Costa's user avatar
7 votes
0 answers
138 views

Explicit framed null bordism realizing $\eta\nu =0$ in stable homotopy group of spheres

There are many standard results in the stable homotopy group of spheres (or equivalently framed bordism groups), about which I would like to acquire better geometric understanding. For example I ...
Yuji Tachikawa's user avatar
4 votes
1 answer
239 views

Two $E_\infty$ structures on infinite matrices

Let $O$ be the infinite orthogonal group. By taking a colimit of the diagram of topological groups $O(1) \to O(2) \to O(3) \to \ldots$, we know $O$ has a continuous group operation given by matrix ...
guest313131's user avatar
3 votes
0 answers
59 views

Example of a factorisation of functors $F = HK$ for which the Kan extension of $F$ along $K$ is not $H$

I was reading Emily Riehl's book: Categorical Homotopy theory, and I encountered exercise 1.1.3: Exercise 1.1.3: Construct a toy example to illustrate that if $F$ factors through $K$ along some ...
julio_es_sui_glace's user avatar
6 votes
1 answer
220 views

Does a complex-oriented $E_1$ ring spectrum (not assumed to have graded-commutative homotopy groups) receive a map from $MU$?

It's well-known that complex cobordism $MU^\ast$ is universal among complex-oriented associative, graded-commutative cohomology theories $E$. This means that if $E$ is a multiplicative cohomology ...
Tim Campion's user avatar
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