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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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Complex $K$-theory of extended powers of a Moore spectrum
Consider a Moore spectrum $S^n/p$. Has the $K$-theory of the extended powers of $S^n/p$ been computed?
For some context: equivalently, I'd like to have the free $E_\infty$-algebra over $KU$ on the $ …
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500
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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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Unobstructedness of braided deformations of symmetric monoidal categories in higher category...
Let $k$ be a field of characteristic zero, and $\mathcal{C}$ be a $k$-linear additive symmetric monoidal category. A braided deformation of $\mathcal{C}$ over a local artin ring $R$ with residue fiel …
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571
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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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573
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Is there a Wall finiteness obstruction in other settings?
Let $\mathcal{S}$ be the $(\infty, 1)$-category of spaces. Then the compact objects of $\mathcal{S}$ are precisely the retracts of finite CW complexes. These are not the same as the finite CW complexe …
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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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$\mathbb{Z}/2$-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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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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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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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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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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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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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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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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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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