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Results tagged with homotopy-theory
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user 102390
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.
5
votes
Accepted
commutative "subalgebras" of associative ring spectra
Let $A$ be an $\mathbf{E}_1$-ring, and let $x\in \pi_n A$. There are two distinct cases to consider. First, if $n = 0$, then the answer to your question is that $x$ is in the image of an $\mathbf{E}_1 …
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5
votes
A question about maps of spectra
If $X$ has homotopy groups in dimensions above $n-1$, then $\mathrm{Map}(X, Y) = \mathrm{Map}(X, \tau_{\geq n} Y)$ for all $Y$. Similarly, if $Y$ has homotopy groups only up through dimension $n$, the …
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9
votes
Is $\mathbb{H}P^\infty_{(p)}$ an H-space?
Sorry for dredging up this question, but here is another argument (at least for $p$ odd, but maybe you don't need this) that came up while thinking about an unrelated problem. If $\mathbf{H}P^\infty_{ …
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11
votes
1
answer
408
views
Finite complexes which are not Thom spectra
I'll be working in the stable world. It's an easy observation that any 2-cell complex (over the sphere) with bottom cell in dimension zero is a Thom spectrum: any such complex is the cofiber of some e …
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6
votes
Accepted
Is the mod-2 Moore spectrum a retract of a shift of its tensor square?
The mod $2$ cohomology of $S^0/2 \wedge S^0/2$ is a $\mathbf{F}_2$-vector space on generators in degrees 0, 1, 1, and 2. The classes in degrees 0 and 2 are connected by a nontrivial $\mathrm{Sq}^2$, s …
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2
votes
Italian-style algebraic geometry in homotopy theory?
This is just a long comment. Homotopy theory is a rather broad field, so the answer to your question depends on what part of homotopy theory you'd like to see having interactions with "Italian-style" …
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6
votes
Accepted
Universal property of $\mathbb S[z]$ and $E_\infty$-ring maps
$\newcommand{\E}{\mathbf{E}}$Dylan answered question 3 (and hence question 1) in the comments, but here's another equivalent way to see it: $\E_\infty$-maps $S^0[z]\to R$ with $R$ a discrete ring (i.e …
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18
votes
2
answers
868
views
Detecting the Brown-Comenetz dualizing spectrum
The Brown-Comenetz dualizing spectrum $I_{\mathbf{Q/Z}}$ is not detected by very many spectra: it is $BP, \mathbf{Z}, \mathbf{F}_2, X(n)$ for $n\geq 2$, and even $I_{\mathbf{Q/Z}}$-acyclic. However, i …
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8
votes
Accepted
The connective $k$-theory cohomology of Eilenberg-MacLane spectra
Charles Rezk already answered this in the comments; I'll just expand on what he wrote. This paper discusses what's now known as Mahowald-Rezk duality; this is a version of Anderson duality that takes …
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3
votes
Calculate homotopy groups of $\mathbb{Z}_2$-equivariant loop spaces of "complex" topological...
$\newcommand{\Z}{\mathbf{Z}}\newcommand{\Map}{\mathrm{Map}}$Let $\sigma$ denote the sign representation of $\Z/2$, and let $S^{d\sigma}$ denote the one-point compactification of $\sigma^{\oplus d}$. L …
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6
votes
1
answer
491
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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11
votes
0
answers
432
views
$E_\infty\mathrm{Spaces}(\mathbf{Z}/p\mathbf{Z},GL_1(E_n))$ and Eilenberg-Maclane spaces
$\newcommand{\Z}{\mathbf{Z}}$Let $p$ be a prime. In his answer here, Jacob Lurie conjectured that $E_\infty\mathrm{Spaces}(\mathbf{Z}/p\mathbf{Z},GL_1(E_n))\simeq K(\Z/p\Z,n)$ where $E_n$ denotes the …
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16
votes
Accepted
The structure of complex cobordism cohomology of the Eilenberg-Maclane spectrum
One can prove that $\mathrm{Map}(H\mathbf{F}_p,MU)$ is contractible. We know that $H\mathbf{F}_p$ is dissonant (Theorem 4.7 of Ravenel's "Localization with Respect to Certain Periodic Homology Theorie …
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2
votes
Accepted
Homology of a loop-suspension space and action of $\mathcal{D}_1$-operad
$\newcommand{\E}{\mathbf{E}} \newcommand{\co}{\mathcal{O}} \newcommand{\free}{\mathrm{Free}} \newcommand{\H}{\mathrm{H}}$Here's one way of seeing the Bott-Samelson theorem. The James splitting gives a …
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10
votes
Accepted
Homotopy coherent generalization of classifying space theory
$\newcommand{\cS}{\mathcal{S}}\newcommand{\Fun}{\mathrm{Fun}}\newcommand{\LMod}{\mathrm{LMod}}\newcommand{\Sp}{\mathrm{Sp}}$Hey Laurent :) Let $X$ be a space (which I'll view as a Kan complex), and le …
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