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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questions
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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)
6
votes
1answer
280 views
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 ...
2
votes
1answer
67 views
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 ...
4
votes
1answer
140 views
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, ...
3
votes
0answers
115 views
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 ...
3
votes
1answer
136 views
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
...
6
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0answers
243 views
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 & \...
3
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1answer
134 views
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 ...
8
votes
1answer
337 views
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 ...
11
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335 views
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:
...
7
votes
0answers
177 views
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 ...
8
votes
1answer
240 views
$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 ...
4
votes
0answers
264 views
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 (...
12
votes
1answer
469 views
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 ...
3
votes
1answer
86 views
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 ...
3
votes
1answer
76 views
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 ...
3
votes
2answers
223 views
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$. (...
5
votes
1answer
132 views
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 ...
17
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0answers
374 views
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 ...
4
votes
1answer
122 views
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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1answer
401 views
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) \...
4
votes
1answer
184 views
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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vote
0answers
66 views
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$ ...
5
votes
1answer
166 views
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 ...
8
votes
1answer
276 views
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 ...
3
votes
1answer
197 views
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 ...
33
votes
3answers
4k views
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 ...
6
votes
1answer
198 views
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 ...
3
votes
1answer
224 views
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 ...
6
votes
0answers
71 views
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 ...
10
votes
1answer
265 views
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 ...
4
votes
0answers
214 views
$L_\infty$-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{...
6
votes
1answer
121 views
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 ...
3
votes
0answers
198 views
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 ...
25
votes
5answers
2k views
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 ...
4
votes
0answers
75 views
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 ...
5
votes
1answer
295 views
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 ...
2
votes
0answers
88 views
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-...
4
votes
0answers
111 views
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 ...
7
votes
1answer
397 views
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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0answers
111 views
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 ...
6
votes
1answer
256 views
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 ...
6
votes
0answers
97 views
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 ...
7
votes
1answer
341 views
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,...
5
votes
0answers
94 views
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 ...
8
votes
2answers
464 views
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 ...
4
votes
0answers
88 views
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 ...
7
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0answers
115 views
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 ...
6
votes
2answers
385 views
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 ...
3
votes
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 ...
