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.
12
votes
3answers
705 views
Does an H-space have at most one delooping?
I am new to H-spaces, delooping, etc. I know that not every $H$-space has a delooping (e.g. Stasheff's theorem, one needs a group-like $A_\infty$ space). I also know that the same space can have ...
10
votes
1answer
251 views
Finite complexes which are not Thom spectra
I'll be working in the stable world. It's an easy observation that any 2-cell complex (over the sphere) with bottom cell in dimension zero is a Thom spectrum: any such complex is the cofiber of some ...
3
votes
0answers
52 views
Unstable and stable looping and delooping
I have some basic questions on the relation of looping and delooping in the stable and unstable homotopy categories. I state them it in the motivic setting, but if somebody has an answer for an ...
-2
votes
1answer
59 views
Alternating property of H_2(T, Z)
Let us consider the torus $T = S^1_X \times S^1_Y$, where the former $S^1_X$ has the coordinate $X$, whereas the latter $S^1_Y$ has $Y$. We have an alternate property $dX \wedge dY = - dY \wedge dX \...
5
votes
1answer
127 views
Quillen equivalent module categories
Let $f:A \rightarrow B$ be a weak equivalence of simplicial commutative rings. There is a Quillen pair $(-\otimes_{A}B, f_{\ast})$ which is an equivalence. In this situation, $(-\otimes_{A}B, f_{\ast})...
9
votes
1answer
305 views
Power operations from a Tate construction
In an action-packed three pages of Lurie's DAG-XIII: Rational and p-adic Homotopy Theory, section 2.2: Power Operations on $\mathbb{E}_{\infty}$-algebras, one finds a construction of the power ...
-1
votes
1answer
141 views
Alternate property of H^2(T, Z) [on hold]
Let us take $T = S^1_X \times S^1_Y$, which is a torus, where the former $S^1_X$ has the coordinate $X$, whereas the latter $S^1_Y$ has $Y$. If we consider the generator $dX \wedge dY \in H^2(T, {\Bbb ...
3
votes
1answer
229 views
Splitting of $H\mathbb{Z}$-module spectra
It is classical result of Adams that every $H\mathbb{Z}$-module spectra splits as a wedge of Eilenberg-MacLane spectra. Let me briefly recall what he writes about the proof.
Let $M$ be an $H\mathbb{Z}...
1
vote
1answer
159 views
The sheafification of taking cohomology is trivial?
Consider the Nisnevich site of a noetherian scheme $S$ of finite Krull dimension (the objects are schemes $U$ smooth and of finite type over $S$), let $A$ be a sheaf of abelian groups on this site. I ...
2
votes
0answers
106 views
$E_\infty$-algebras and Tor-unital rings
Recall that a non-unital ring $R$ is called Tor-unital if $Tor^1_{R_+}(\mathbb Z,\mathbb Z) \cong 0$ where $R_+$ is the unitalization of $R$. See e.g. https://arxiv.org/pdf/1610.04998.pdf. If $R$ is ...
4
votes
1answer
191 views
Link between homotopy equivalence of simplicial sets and categorical equivalences
In Higher Topos Theory, a map $f: S \rightarrow T$ of simplicial sets is a categorical equivalence if after applying the functor $\mathfrak{C}[-]$ we have an equivalence of simplicial categories.
In ...
11
votes
1answer
689 views
The homology of the orbit space
Suppose we have an acyclic group $G$ and let $X$ be a contractible CW-complex such that $G$ acts freely on $X$ (we do not suppose that the action is proper).
Is there a way to understand the homology ...
23
votes
1answer
346 views
Modern survey of unstable homotopy groups?
Toda no doubt made some big strides when computing unstable homotopy groups $\pi_{n+k}(S^n)$ for $k < 20$ which his collaborators later improved upon.
The methods he used are documented in his ...
5
votes
2answers
220 views
Limit of weak equivalences in a Bousfield localization
Let $M$ be a model category and $C$ a class of maps in it, and assume the left Bousfield localization $L_CM$ exists. Suppose we are given sequences of maps $(p_{n+1}: X_{n+1}\to X_n), (q_{n+1}: Y_{n+1}...
6
votes
1answer
158 views
Precise reference for the equivalence of $E_n$ algebras and locally constant factorization algebra?
I've seen the following theorem attributed to Lurie:
Theorem. There is an equivalence of $(\infty,1)$-categories between $E_n$ algebras and locally constant factorization algebra on $\mathbf{R}^n$.
...
2
votes
0answers
92 views
Is there a definition of an unpointed schematic homotopy type?
In the paper Champs Affines (http://www.math.univ-toulouse.fr/~btoen/chaff.pdf) Toen introduces pointed schematic homotopy types (SHTs) to solve Grothendieck's schematization problem (described in ...
5
votes
0answers
119 views
DG-Modules over CDG-algebras in the sense of rational homotopy theory
I don't know if this question is elementary or not. Suppose we have a rationalization $X_\mathbb{Q}$ of a simply connected topological space $X$. Then we can construct a CDG-algebra - a minimal ...
1
vote
0answers
101 views
Whitehead Theorem for maps
Let us consider two simply-connected CW complexes. Combining the theorems of Whitehead and Hurewicz we have that a map between them is an equivalence if and only if its induced map on integral chains ...
2
votes
0answers
60 views
resolution of differential graded algebras.
Suppose that we have tree maps of differential graded algebras
$A\rightarrow B$, $A\rightarrow C$ and $A\rightarrow D$ such taht $A\rightarrow C$ is a trivial cofibration of differential graded ...
16
votes
3answers
787 views
Are there prominent examples of operads in schemes?
There is an abundance of examples of operads in topological spaces, chain complexes, and simplicial sets. However, there are very few (if any) examples of operads in algebraic geometric objects, even ...
13
votes
0answers
322 views
How well-defined is $\bar\kappa$ in the stable $20$-stem?
The $2$-completed stable $20$-stem $\pi_{20}(S)_2$ is cyclic of order $8$.
Mimura and Toda (1963, Lemma 15.4) mr=157384 show the existence of a class $\bar\kappa_7 \in \pi_{27}(S^7)$ whose stable ...
3
votes
0answers
133 views
filetered colimit of fibrant-cofibrant objects
Suppose that we have a $\lambda$-combinatorial model category $M$ (for some cardinal $\lambda$) such that any $\lambda$-filtered colimit of fibrant-cofibrant objects is always fibrant. My question is ...
5
votes
1answer
78 views
On the existence of a domination map of a finite polyhedron
A continuous map $d:X\to A$ is called domination if there exists a map $u:A\to X$ so that $d\circ u\simeq 1_A$.
Is there a domination map $d:P\to P$ of a finite polyhedron $P$ so that $d$ is not a ...
14
votes
1answer
321 views
Homotopy fixed points of complex conjugation on $BU(n)$
Stably it is known that $(\mathbb Z\times BU)^{hC_2}\simeq \mathbb Z\times BO$ holds. The homotopy fixed point spectral sequence for KU with complex conjugation action can be completely calculated and ...
11
votes
1answer
401 views
homotopy and (co)filtered limits
Suppose we have a (co)filtered digaram $\dots \rightarrow X_{2}\rightarrow X_{1}$ of topological space. Is is true that the natural map $\pi_{0}[\lim X_{i}]\rightarrow \lim \pi_{0}(X_{i})$ is an ...
5
votes
0answers
135 views
h-principle for pairs
Let $A,B$ be complex analytic spaces. Suppose that $[A,B]$ satisfies the h-principle: i.e. every class of continuous function $f:A \to B$ up to homotopy, contains a holomorphic representative. Let $C \...
3
votes
1answer
100 views
Making immersions from immersion conjecture into embeddings
Is it true that any smooth manifold of dimension $n$ can be smoothly embedded into $\mathbb{R}^{2n+1-a(n)}$ where $a(n)$ is the number of appearances of digit "1" in the binary expansion of $n$?
6
votes
1answer
381 views
Unstable Greek letter elements
A theorem of Hopkins and Mahowald states that the Thom spectrum of the map $\Omega^2 S^3 \to B\mathrm{GL}_1(\mathbb{S}_{(p)})$ classifying the element $p$ is exactly $\mathrm{H}\mathbf{F}_p$. Let $T(1)...
7
votes
0answers
256 views
What is this analog of $\mathbb A^1$ / proper homotopy theory?
In homotopy theory, we can construct the $\infty$-category of spaces from the ordinary category of oriented manifolds $\rm Man$, by freely co-completing it, imposing gluing relations, and homotopy ...
4
votes
1answer
191 views
A finite Whitehead Theorem for $\infty$-topos
Let's consider in an $\infty$-topos, we have an object $X$ of homotopy dimension $\leq n$ (in the sense of Lurie HTT), let $f: A\to B$ be an $n$-equivalence morphism. Can we conclude that $f$ induces ...
8
votes
1answer
193 views
How is topological André-Quillen homology (TAQ) a “stabilization”, exactly?
Let $S\to A\to B$ be cofibrations of commutative $S$-algebras. Then the topological André-Quillen $B$-module $TAQ(B|A)$ can be computed as a stabilization. Precisely, I think it means the following: ...
8
votes
1answer
415 views
Formality over $\mathbb{R}$ vs formality over $\mathbb{Q}$
On ncatlab page on formality, it is stated that Deligne--Griffiths--Morgan--Sullivan proved that the real homotopy type of a closed Kaehler manifold is formal. Later, Sullivan "improved" this to $\...
4
votes
0answers
268 views
Quillen pair, fibrant-cofibrant objects
This question is a follow-up of the question I have asked today : left quillen functor and fibrant objects
Suppose that we have $$ L :C\leftrightarrow D: R$$
an adjoint Quillen pair. We assume that ...
1
vote
0answers
75 views
Strict units in A-infinity algebras
In Kontsevich-Soibelman's paper "Notes on $A_\infty$-algebras, $A_\infty$-categories and non-commutative geometry", $A_\infty$-algebras with strict units are defined so units act trivially on higher ...
32
votes
2answers
3k views
Why is Voevodsky's motivic homotopy theory 'the right' approach?
Morel and Voevoedsky developed what is now called motivic homotopy theory, which aims to apply techniques of algebraic topology to algebraic varieties and, more generally, to schemes. A simple way of ...
3
votes
0answers
182 views
Can a functorial factorization be modified so that it fixes the initial object?
Consider a category $\mathcal C$ with a weak factorization system which is functorial. Let $*$ be an initial object. If $X\in \mathcal C$, denote by $\eta_X:*\to X$ the unique map. Using the given wfs,...
3
votes
0answers
54 views
Monoidal equivalence of categories of modules in different models of higher algebra
A result of Shipley states the following (in the language of model categories):
For a differential graded algebra $A$, the $\infty$-category of (dg)-modules over $A$ agrees with the $\infty$-category ...
9
votes
1answer
478 views
The structure of complex cobordism cohomology of the Eilenberg-Maclane spectrum
Let $MU$ be the complex bordism spectrum and let $H\mathbb{Z}$ be the Eilenberg-Maclane spectrum.
Is it know what the structure of the complex cobordism cohomology $MU^{*}(H\mathbb{Z})$ is?
EDIT: ...
10
votes
1answer
334 views
Whitehead products in homotopy groups of spheres
Here is what I know about Whitehead products in homotopy groups of spheres:
$[\mathrm{id}_{S^{2n}},\mathrm{id}_{S^{2n}}]$ has Hopf invariant (EDIT: $\pm$) two.
No element that survives into the ...
7
votes
1answer
319 views
Reference request for K-Theory linearization
I posted this question on math.se here:https://math.stackexchange.com/q/2996787/482732, but I think it may be more appropriate here, sorry if I am wrong about that.
In Waldhausen's paper Algebraic K ...
2
votes
1answer
90 views
$M$ is a manifold and isometrically embedded in $X$, homotopy type of $M$ is determined by polyhedrons $P$ s.t. $M\subseteq P \subseteq X$?
This is the setting.
$M$ is a compact, connected Riemannian manifold without boundary. and it is isometrically embedded in some larger metric space $X$ ($X$ is not necessarily manifold). So, one can ...
5
votes
2answers
184 views
Is the underlying vector space of the minimal model of an $A_{\infty}$-algebra canonical?
On the page 4 of these notes it is stated that an $A_{\infty}$-algebra $A$ is necessarily is quasi-isomorphic to an $A_{\infty}$-algebra $HA$ with trivial differential. Moreover, $HA$ is unique up to ...
2
votes
1answer
160 views
Localization of a model category with respect to a class of maps
I am little bit lost with the following (standard?) problem in model categories.
Suppose we have a Quillen adjunction between combinatorial model categories:
$$L:M\leftrightarrow N: R $$
and let $(...
11
votes
1answer
361 views
Is there an expository account of homology of simplicial sets that does not assume prior familiarity with any variant of homology?
There are numerous expositions of simplicial homology in the literature.
Munkres in “Elements of Algebraic Topology” develops the homology theory of simplicial complexes.
Hatcher in “Algebraic ...
13
votes
1answer
558 views
Why $K(X) \longrightarrow G (X)$ is a Poincaré duality for K-theory?
It's well known that for Noetherian separated regular schemes the canonical map $$K(X) \longrightarrow G(X)$$ (Quillen uses $K'$ instead of $G$, though) is a weak equivalence.
This statement is ...
14
votes
1answer
413 views
Spectra with “finite” homology and homotopy
As known, any non-trivial finite spectrum $X$ can not have non-zero homotopy groups $\pi_i(X)$ only for finite number of $i$. As I understand, the same is true for any spectrum $X$ with finitely ...
6
votes
1answer
167 views
HKR generalized character theory question regarding a certain construction
In that paper https://web.math.rochester.edu/people/faculty/doug/mypapers/hkr.pdf Hopkins-Kuhn-Ravenel introduced the idea of generalized character corresponding to a complex-oriented cohomology ...
1
vote
0answers
40 views
Maps having the right lifting property against cofibrations of compact spaces
I would like to know the properties of the maps that have the right lifting property against cofibrations of compact spaces. By definition, they are acyclic Serre fibrations, but I would hope to be ...
5
votes
0answers
143 views
Homotopy functor calculus vs functor calculus in additive categories
Consider the Goodwillie calculus of a homotopy functor $F : \mathrm{Sp} \to \mathrm{Sp}$, where $\mathrm{Sp}$ denotes an appropriate model for spectra (... orthogonal spectra for instance).
Then ...
12
votes
2answers
323 views
Is there any significance to Bousfield localization in the non-derived context?
The term "Bousfield localization" of a category $C$ is used in roughly two different ways:
There is a general usage (as in model categories or triangulated categories), which $\infty$-categorically ...
