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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 ...

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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
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1answer
74 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 ...
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299 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 ...
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1answer
388 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 ...
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93 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 \...
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1answer
93 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
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1answer
369 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)...
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251 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 ...
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1answer
181 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
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1answer
183 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: ...
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1answer
395 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 $\...
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221 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 ...
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73 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 ...
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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 ...
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177 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,...
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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
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1answer
468 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: ...
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1answer
325 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
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1answer
317 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 ...
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1answer
88 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 ...
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170 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 ...
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1answer
158 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 $(...
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1answer
351 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 ...
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1answer
532 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 ...
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1answer
399 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 ...
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1answer
164 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 ...
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37 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
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137 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
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2answers
318 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 ...
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1answer
222 views

How to define 0-sphere in a category with zero object?

The 0-sphere $S^0$ is the coproduct of two points, $$S^0 \simeq \ast \coprod \ast$$ How to define 0-sphere in a category with zero object? Let $\mathcal{C}$ be a category. A cylinder, $\mathbf{I}$, ...
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2answers
285 views

Maps from 2-Torus to SO(3)

Could someone please point me to a reference for topologically nontrivial maps from 2-Torus to SO(3), and how they are classified? [I'm a physicist, so a simple explanation would be useful]
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2answers
183 views

Weak homotopy equivalence between $\Omega \underset{\rightarrow}{\lim}Z_n$ and $\underset{\rightarrow}{\lim}\Omega Z_n$

Let $Z_1 \rightarrow Z_2 \rightarrow\cdots$ be an arbitrary sequence of CW-complexes and let $\Omega X$ denote the loop space over $X$. In Allen Hatcher's "Algebraic Topology" (http://pi.math.cornell....
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2answers
312 views

Do finite simplicial sets jointly detect isomorphisms in the homotopy category? [duplicate]

Let $\mathcal{H}$ denote the homotopy category associated with the Kan-Quillen model structure on $\mathbf{sSet}$. Suppose we have a map $f\colon X \to Y$ between Kan complexes, such that for every ...
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2answers
454 views

Sphere spectrum, Character dual and Anderson dual

The homotopy groups of the sphere spectrum are the stable homotopy groups of spheres. However, could you help me to appreciate the mathematical meanings of the following: What is the significance of ...
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77 views

Axiomatization of the shuffle decomposition

I am trying to figure out an axiomatization for Reedy categories such that the product of representables admits a shuffle decomposition. Cisinski suggested imposing a condition, which we will state ...
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1answer
249 views

Commutativity up to homotopy implies strict commutativity, for lifting problems

Suppose we have a commutative diagram $\require{AMScd}$ \begin{CD} A @>>> X \\ @VVV & @VVV \\ W @>>> Y\\ \end{CD} where the map $A\rightarrow W$ is a cofibration and the ...
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1answer
301 views

Milnor Conjecture on Lie groups for Morava K-theory

A conjecture by Milnor state that if $G$ is a Lie group, then the map $B(G^{disc})\to BG$ sending the classifying space of $G$ endowed with the discrete topology to the classifying space of the ...
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356 views

Questions about obstruction theory (Hatcher's book)

I'm actually studying obstruction theory as presented in the last section of chapter $4$ of the book Algebraic Topology by Allen Hatcher. He first finds condition so that a space $X$ admits a ...
2
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1answer
202 views

A question on eversion of (odd) spheres

At the right column of the page 654 of the paper, R. Palais, The Visualization of Mathematics: Towards a mathematical Exploratorium, Notice AMS it is written "There can be no eversion of ...
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Is there a simple algebraic setup to accomodate fibres and cofibres at the same time?

If I understand it correctly, there are two mutually dual "leading principles" in homotopy theory: never perform quotients, add structure instead; never require subobjects, take fibres instead. ...
6
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2answers
252 views

Critical points and high homotopy groups

Is there any known or interesting relation between critical points (possibly degenerate, or maybe only nondegenerate) of a function on a manifold and generators/relations of high homotopy groups? I ...
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0answers
40 views

sub relative cell complex

This is a question concerning the proposition 11.4.8 of P.Hirschhorn's book Model categories and their localizations. The proposition 11.4.8 is an analogous, in my opinion, to the well known ...
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1answer
96 views

Naturality of minimal model of a fibre bundle

$\require{AMScd}$ For rational fibrations $F \rightarrow E \rightarrow B$, we have a nice description in terms of sullivan minimal models, namely a commutative diagram of cdga's $$ \begin{CD} ...
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218 views

Dualizable objects in homotopy category of chain complexes

The proposition 1.9 from "Duality, Trace and Transfer" by Dold and Puppe states that: Given a commutative ring $R$, a chain complex of $R$-modules is strongly dualizable in $Ho(Ch(R))$, the homotopy ...
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240 views

Is there a citeable source for generators and relations of simplicial sets?

Simplicial sets can be specified using generators and relations, in complete analogy with groups, rings, etc. More precisely, a system of generators of relations for a simplicial set consists of a ...
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238 views

“Complementarity” between homotopy and cohomology [duplicate]

I was browsing MO and I have stumbled upon this answer which discusses why we should expect homotopy groups of spheres to be complicated. One heuristic argument given is that "the theory needs to have ...
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114 views

Known obstruction for efficient computation of Stable homotopy groups?

Computation of stable homotopy groups (for example of sphere) is hard, but still, not as hard as unstable ones. For unstable homotopy groups there are some results showing that there cannot be ...
4
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1answer
178 views

Conceptual and practical reasons and consequences of inverting weak equivalences

Although dealing with this in one or other form for many years, to my shame this question only struck me now. One of the most radical differences between categories of "algebraic" and "topological" ...
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205 views

Did the Goerss-Hopkins manuscript “Multiplicative stable homotopy theory” ever appear?

A citation to "M. J. Hopkins and P. Goerss, Multiplicative stable homotopy theory, unpublished manuscript, 1996" appears in the Hill, Hopkins, Ravenel Annals paper on the Kervaire invariant. It was ...
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1answer
408 views

Is there an explicit Dold-Thom theorem?

The Dold-Thom theorem tells us that we can recover the reduced homology of a pointed space $(X,x)$ via taking homotopy groups of the symmetric product: $$\pi_i(\mathrm{Sym}^{\infty}(X,x)) \cong H_i(X,...