# 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 ...

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### 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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### 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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### 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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### 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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### 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 $n$?

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### 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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### 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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### 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 ...

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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...

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### 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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### 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 ...

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### 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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### $M$ 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...

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### 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 ...

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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...

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### 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.
...

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### 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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### 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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### 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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### 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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### 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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### “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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### 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 ...

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### 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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### 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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### 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,...