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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Which finite abelian groups aren't homotopy groups of spheres?

Someone asked me if all finite abelian groups arise as homotopy groups of spheres. I strongly doubted it, and I bet ten bucks that $\mathbb{Z}_5$ is not $\pi_k(S^n)$ for any $n,k$. But I don't know ...
John Baez's user avatar
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Cochains on Eilenberg-MacLane Spaces

Let $p$ be a prime number, let $k$ be a commutative ring in which $p=0$, and let $X = K( {\mathbb Z}/p {\mathbb Z}, n)$ be an Eilenberg-MacLane space. Let $F$ be the free $E_{\infty}$-algebra over $k$ ...
Jacob Lurie's user avatar
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Is there anything to the obvious analogy between Joyal's combinatorial species and Goodwillie calculus?

Combinatorial species and calculus of functors both take the viewpoint that many interesting functors can be expanded in a kind of Taylor series. Many operations familiar from actual calculus can be ...
Tim Campion's user avatar
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Homotopy type of TOP(4)/PL(4)

It is known (e.g. the Kirby-Siebenmann book) that $\mathrm{TOP}(n)/\mathrm{PL}(n)\simeq K({\mathbb Z}/2,3)$ for $n>4$. I believe it is also known (Freedman-Quinn) that $\mathrm{TOP}(4)/\mathrm{PL}(...
Ricardo Andrade's user avatar
36 votes
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Functor that maps to both $KO^n$ and $KO^{-n}$

(my question is also meaningful for complex K-theory, but since Kn(X) is always isomorphic to K-n(X), it's less interesting) I start by recalling the analytic definition of KO-theory: The following ...
André Henriques's user avatar
33 votes
0 answers
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Is there a (discrete) monoid M injecting into its group completion G for which BM is not homotopy equivalent to BG?

For a (discrete) monoid $M$, the classifying space $BM$ is the geometric realization of the nerve of the one object category whose hom-set is $M$. (This definition gives the usual classfiying space ...
Omar Antolín-Camarena's user avatar
31 votes
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834 views

The central insight in the proof of the existence of a class of Kervaire invariant one in dimension 126

I understand from a helpful earlier MO question that the techniques leading to the celebrated resolution of the Kervaire invariant one problem in the other candidate dimensions yield no insight on ...
jdc's user avatar
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28 votes
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On the (derived) dual to the James construction.

Background If $X$ is a based space then the James construction on $X$ is the space $J(X)$ given by $$ X \quad \cup \quad X^{\times 2} \quad \cup \quad X^{\times 3} \quad \cup \quad \cdots $$ in ...
27 votes
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Spectral sequences as deformation theory

I believe that running the spectral sequence of a filtered complex / spectrum $ \cdots \to F_n \to F_{n+1} \to \cdots$ can be viewed as doing deformation theory in some very primitive "derived ...
Tim Campion's user avatar
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25 votes
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Chromatic Spectra and Cobordism

I apologize in advance, if some of the things I've written are incorrect. The cobordism hypothesis states that $\mathbf{Bord}^\mathrm{fr}_n$ is the free symmetric monoidal $(\infty,n)$-category with ...
Nerses Aramian's user avatar
21 votes
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705 views

If $X\times Y$ is homotopy equivalent to a finite-dimensional CW Complex, are $X$ and $Y$ as well?

Is there a space $X$ that is not homotopy equivalent to a finite-dimensional CW complex for which there exists a space $Y$ such that the product space $X\times Y$ is homotopy equivalent to a finite-...
David Sykes's user avatar
20 votes
0 answers
344 views

Homotopic version of Freyd's AT category observations

Freyd was the first to formalize a striking comparison between abelian categories and topoi, showing that their exactness properties can be jointly captured by the axioms of AT categories, and the ...
Mathemologist's user avatar
19 votes
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622 views

Homotopy type of the affine Grassmannian and of the Beilinson-Drinfeld Grassmannian

The affine Grassmannian of a complex reductive group $G$ (for simplicity one can assume $G=GL_n$) admits the structure of a complex topological space. More precisely, the functor $$X\mapsto |X^{an}|$$ ...
W. Rether's user avatar
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Eckmann-Hilton argument / Grothendieck

In terse terms, the “Eckmann-Hilton argument” says that a monoid in the category of monoids is commutative. It is essentially used, for example, to prove that homotopy groups of topological groups are ...
ACL's user avatar
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19 votes
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501 views

Other examples of computations using transfer of structure from the chains to the homology?

There is a `long' history of transfer (up to homotopy!) of algebraic structure from a dg _ algebra A to its homology H(A) (e.g. Kadeishvili for the associative case and Heubschmann for the Lie case). ...
Jim Stasheff's user avatar
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18 votes
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About the equivariant analogue of $G_n/O_n$

Let $BO_n$ and $BG_n$ be the classifying spaces for rank $n$ vector bundles and for spherical fibrations with fiber $S^{n-1}$, respectively, and let $G_n/O_n$ be the homotopy fiber of $BO_n\to BG_n$. ...
Tom Goodwillie's user avatar
18 votes
0 answers
444 views

Orientation-reversing homotopy equivalence

If there is an orientation-reversing homotopy equivalence on a closed simply-connected orientable manifold is there an orientation-reversing homeomorphism? It is not true, for instance, that if there ...
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18 votes
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642 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 ...
Gregory Arone's user avatar
18 votes
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685 views

Homotopy groups of spheres and differential forms

The only infinite homotopy groups of spheres are $\pi_n(\mathbb{S}^n)$ and $\pi_{4n-1}(\mathbb{S}^{2n})$. This is a well known result of Serre. In both cases the nontriviality of these groups can be ...
Piotr Hajlasz's user avatar
18 votes
0 answers
506 views

Does the "holomorphic spheres-to-continuous spheres" forgetful function respect the mixed Hodge structures on homotopy groups?

For each smooth, projective, complex variety $X$ that is simply connected, John Morgan constructed a natural mixed Hodge structure on the homotopy group $\pi_k(X,x)\otimes \mathbb{Q}$. This was ...
Jason Starr's user avatar
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18 votes
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501 views

Lipschitz constant of a homotopy

Let $n>0$, $\mathbb S^n$ be $n$-sphere and $1\in \mathbb S^n$ be its north pole. A am looking for an example of compact manifold $M$ with a continuous $n$-parameter family of maps $h_x\colon M\to ...
Anton Petrunin's user avatar
17 votes
0 answers
654 views

Proof of MacPherson's result about set-valued constructible sheaves and exit paths

I'm looking for a proof of a theorem that is attributed to MacPherson. Treumann (Section 1.1 in Exit paths and constructible stacks, 2009) states the theorem as: Theorem 1.2 (MacPherson). Let $(X,S)...
Jānis Lazovskis's user avatar
17 votes
0 answers
380 views

Kan's simplicial formula for the Whitehead product

In his article on Simplicial Homotopy Theory (Advances in Math., 6, (1971), 107 –209) Curtis quotes a formula (on page 197) for the Whitehead and Samelson products in a simplicial group $G$. The ...
Tim Porter's user avatar
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17 votes
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719 views

The spectral sequence of a tower of principal fibrations

Assume we have a tower of fibrations (of simplicial sets, let's say): $$\cdots\rightarrow X_{n+1}\rightarrow X_n\rightarrow\cdots\rightarrow X_0.$$ Let $X=\lim_nX_n$ be the (homotopy) inverse limit. ...
Fernando Muro's user avatar
16 votes
0 answers
272 views

Fibrations whose total spaces are more highly connected than their fibers

The (generalized) Hopf fibrations $S^1 \to S^3 \to S^2$, $S^3 \to S^7 \to S^4$ and $S^7 \to S^{15} \to S^8$ have the property that their total spaces are more highly connected than their fibers. Are ...
Jens Reinhold's user avatar
16 votes
1 answer
2k 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 ...
GSM's user avatar
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16 votes
1 answer
396 views

Embedding of two-dimensional CW complexes which induces a zero homomorphism on second homotopy groups

I am interested in the following question which I asked at MSE and did not get any clues. How to find three two-dimensional CW complexes $K_1, K_2, K_3$ with non-trivial $\pi_2$ and injective ...
Samarkand's user avatar
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15 votes
0 answers
272 views

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
15 votes
0 answers
515 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 ...
John Rognes's user avatar
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669 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 ...
Dmitri Pavlov's user avatar
15 votes
0 answers
520 views

Is this an $E_\infty$-algebra?

I have a particular kind of algebraic structure that's come up in my work. It's basically a chain complex equipped with a multiplication which is commutative and associative up to homotopy in a ...
Dan Petersen's user avatar
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15 votes
0 answers
699 views

Solving polynomial systems with homotopy. Where is the bottleneck?

I have a polynomial system with $n+k$ unknowns ($n+k$ can be greater than 8), that is known to have a limited number of isolated solutions. I want to solve this system numerically, but if I plug it ...
Ulderique Demoitre's user avatar
14 votes
0 answers
306 views

Do Sullivan's and Wilkerson's algebraic group structures on rational homotopy automorphisms coincide?

For a 1-connected space $X$ with the homotopy type of a finite CW-complex, the homotopy automorphisms $\pi_0\,\mathrm{hAut}(X)$ are known to have the structure of an arithmetic group up to finite ...
skupers's user avatar
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14 votes
0 answers
289 views

Which limits distribute over which colimits in $Set$? How about in $Spaces$?

I've never really thought much about distributivity of limits and colimits -- I tend to think more about commutativity of limits and colimits. This question makes me want to change that. The question ...
Tim Campion's user avatar
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14 votes
0 answers
299 views

Does virtual Morava K-theory have an Eilenberg-Moore spectral sequence?

In a recent question, Tim Campion was interested in analyzing the Morava $K$–theory of a space $X$ by dissecting the space into connective and coconnective parts: $$X(m, \infty) \to X \to X[0, m].$$ ...
Eric Peterson's user avatar
14 votes
0 answers
366 views

When is a map of topological spaces homotopy equivalent to an algebraic map?

My question is simple, but I don't expect there are any simple answers. Let $X$ and $Y$ be a pair of schemes, and let $X(\mathbb{C})$ and $Y(\mathbb{C})$ denote their respective spaces complex points....
Patrick Elliott's user avatar
14 votes
0 answers
237 views

How many cells are needed in a simplicial structure of $\mathbb{S}^n$ to induce all of $\pi_n(\mathbb{S}^m)$

Serre proved, that for (allmost) all $n,m\in\mathbb{N}$ the homotopy groups $\pi_n(\mathbb{S}^m)$ are finite, so - using simplicial approximation - for $n, m$ fixed there is a finite cell ...
Takirion's user avatar
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14 votes
0 answers
784 views

What does convergence of a Bousfield-Kan spectral sequence say about the homotopy type of the totalization?

Given a cosimplicial space or spectrum $X^\bullet$, there is an associated Bousfield-Kan spectral sequence. This starts out as the bigraded object obtained by taking homotopy groups of each $X^n$ and ...
Jonathan Beardsley's user avatar
14 votes
0 answers
711 views

Homological algebra is linearized homotopical algebra?

I have stumbled across statements like Homological algebra is linearized homotopical algebra. Chain complexes are linearizations of simplicial complexes. The Dold-Kan correspondence was often ...
Exterior's user avatar
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14 votes
0 answers
2k views

Homotopy groups of the orthogonal group

I'm interested in knowing what $n$-dimensional vector bundles on the $n$-sphere look like, or equivalently in determining $\pi_{n-1}(SO(n))$; here's a case that I haven't been able to solve. Let $n \...
Akhil Mathew's user avatar
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13 votes
0 answers
225 views

What is the relationship between Goodwillie calculus and derived deformation theory?

Goodwillie calculus is a way of understanding a functor $F$ in terms of its Goodwillie tower, a tower whose limit approximates $F$, whose layers can be understood in terms of stable data. Derived ...
Tim Campion's user avatar
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13 votes
0 answers
289 views

The first two $k$-invariants of $\mathrm{pic}(KU)$ and $\mathrm{pic}(KO)$

$\DeclareMathOperator\Pic{Pic}\DeclareMathOperator\pic{pic}$Real and complex topological $K$-theories, $KO$ and $KU$, have Picard spectra $\pic(KO)$ and $\pic(KU)$ built from the $\mathbb{E}_\infty$-...
Jonathan Beardsley's user avatar
13 votes
0 answers
908 views

Manifolds up to homeomorphisms VS Manifolds up to homotopy equivalence

Let $\mathcal{Man}$ be the set of all connected closed orientable manifolds up to homeomorphism. It is an abelian semigroup with respect to the cartesian product $\times$ (multiplication) and unit $\...
Paris's user avatar
  • 707
13 votes
0 answers
439 views

Structures between PL and smooth

Let $X$ be a topological manifold of dimension at least five. The Kirby-Siebenmann invariant $ks(X)\in H^4(X,\mathbb{Z}_2)$ is an obstruction to the existence of a PL structure on $X$. If it vanishes, ...
Philip Engel's user avatar
  • 1,493
13 votes
0 answers
319 views

One periodic cohomology theories?

Is there a classification of one periodic cohomology theories? In other words, spaces $X$ such that $\Omega X \simeq X$? For example, $X=*$ and $X= K(G,0) \times K(G,1) \times \dots$ are examples. ...
Connor Malin's user avatar
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13 votes
0 answers
393 views

Computations using "Stover's spectral sequence"

In this article from 1990, Stover describes a specral sequence which converges to the higher homotopy groups of the homotopy colimit of a diagram $\underline{X}$ of topological spaces. The second ...
Matt's user avatar
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13 votes
0 answers
257 views

dual to Hodge theory

Let $(M,g)$ be a closed Riemannian manifold. In my understanding Hodge theory shows that any de Rham cohomology class can be represented uniquely by a harmonic form. Moreover the harmonic form ...
Thomas Rot's user avatar
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13 votes
0 answers
847 views

About maps inducing bijections on homotopy classes

Let us assume that $f:X \to Y$ is a map of connected CW complexes, having the following property: if $K$ is a finite CW complex, then the induced map $f_{\ast}:[K,X] \to [K,Y]$ on \emph{free} homotopy ...
Johannes Ebert's user avatar
13 votes
1 answer
465 views

Is the operadic nerve functor an equivalence of ∞-categories?

It is now known that the $\infty$-category of $\infty$-operads as defined by Lurie is equivalent to the underlying $\infty$-category of the model category of simplicial operads, see http://arxiv.org/...
Yonatan Harpaz's user avatar
12 votes
0 answers
444 views

Proofs of Serre's theorem on simply-connected finite CW complexes

A famous result due to Serre states that any simply-connected finite CW complex with non-trivial $\mathbb{Z}_2$ homology has infinitely many non-zero homotopy groups. (In fact, Serre proves more than ...
homotopy-enthusiast's user avatar

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