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Results tagged with homotopy-theory
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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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$\mathcal{I}$-functors and infinite loop spaces
By the Barratt-Priddy-Quillen theorem, the space $B \Sigma_\infty^+$ is the infinite loop space $\Omega^\infty \Sigma^\infty S^0$. I'm curious about a "high-concept" reason that $B \Sigma_\infty^+$ (a …
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11
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Generating acyclic cofibrations for the various model structures in higher category theory
There are a number of model categories important in higher category theory, which provide a "presentation" of some $\infty$-category of $\infty$-categories. For example:
The Joyal model structure on …
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What is the homotopy type of a free simplicial ring?
Is there a good description of the homotopy type of a free simplicial ring (or simplicial $R$-algebra) on a given simplicial set, in terms of the homotopy type of that simplicial set?
(This is mostly …
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2
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$\mathbb{Z}/2$-action on spectra given by inversion
This is a somewhat naive question to which I don't know the answer. There is a map of spectra $S \to S$, defined up to homotopy, given by multiplication by $-1$, and it satisfies the relation $(-1)^2 …
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Finite spectrum annihilated by multiplication by two
Let $X$ be a finite spectrum. Say that $X$ has characteristic two if multiplication by two on $X$ is nullhomotopic.
Does there exist a noncontractible finite spectrum of characteristic two?
(This …
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11
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3
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A homotopyish Landweber exact functor theorem
Let $M$ be a $\pi_*(MU)$-module. The Landweber exact functor theorem gives conditions for the functor that sends a space $X$ to $ MU(X) \otimes_{\pi_*(MU)} M$ to define a homology theory on spaces, wh …
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Homology of homotopy fixed point spectra
Let $E$ be a spectrum acted upon a finite group $G$. Is there a general way of computing the homology of the homotopy fixed point spectrum $E^{hG}$ in terms of that of $E$? (I'm aware that there is a …
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16
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A model category of spaces where strict commutative monoids are $E_\infty$-spaces
There are various strict monoidal model categories of spectra (e.g. symmetric spectra) where the honestly commutative monoid objects model the "coherently commutative" ring spectra (which might otherw …
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Strictly commutative elements of $E_\infty$-spaces
Let $X$ be an $E_\infty$-space (not necessarily grouplike). Let $x \in \pi_0 X$ be an element; say that $x$ is strictly commutative if there is a map of $E_\infty$-spaces $\mathbb{Z}_{\geq 0} \to X$ t …
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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 …
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23
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A homotopy commutative diagram that cannot be strictified
By a "homotopy commutative diagram," I mean a functor $F: \mathcal{I} \to \mathrm{Ho}(\mathrm{Top})$ to the homotopy category of spaces. By a "strictification," I mean a lifting of such a functor to t …
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101
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Is there a high-concept explanation for why "simplicial" leads to "homotopy-theoretic"?
My (limited) understanding is that simplicial methods tend to be used whenever you want some kind of nontrivial homotopy theory -- for instance, to get a nice model structure, you use simplicial sets …
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Geometric interpretation of families in the stable homotopy groups of spheres
There are infinite families in the stable homotopy groups of spheres; many of these can be seen by looking for "periodicity" phenomena in the Adams-Novikov spectral sequence. An example is the image …
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35
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What do loop groups and von Neumann algebras have to do with elliptic cohomology?
Recall that complex $K$-theory is a cohomology theory on topological spaces, which can be described in several equivalent ways:
Given a finite complex $X$, $K^0(X)$ is the Grothendieck group of vec …
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Failure of "equivariant triangulation" for finite complexes equipped with a $G$-action
Let $\mathcal{S}$ be the $\infty$-category of spaces, and let $G$ be a finite group, and let $BG$ be the groupoid with one object and automorphisms given by $G$.
Consider the $\infty$-category $\math …
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