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Results tagged with homotopy-theory
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user 11546
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.
34
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Surveys of Goodwillie Calculus
Is there a good general introduction to Goodwillie calculus out there, like a paper or publication that gives a general overview of the calculus as well as how it is useful and why we are interested i …
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15
votes
2
answers
920
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What homotopy classes can attaching an $E_n$-cell kill?
Let $A$ be a connected $E_{n+1}$-ring spectrum and let $\alpha\in\pi_k(A)$. I am having trouble showing that attaching an $E_n$-cell along $\alpha$ will necessarily not kill an element $\beta\in\pi_k( …
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14
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What does convergence of a Bousfield-Kan spectral sequence say about the homotopy type of th...
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 …
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14
votes
2
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Difficulties with the mod 2 Moore Spectrum
I have been informed that there is a reference out there which specifically details what goes wrong with the mod 2 Moore spectrum, i.e. why it is not $A_\infty$ or something? I do not know the details …
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14
votes
1
answer
348
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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 $\mathbb{E}_\infty$-s …
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12
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2
answers
558
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Why does $Mf$ always support an $Mf$-orientation?
Let $f:X\to BGL_1(\mathbb{S})$ be a morphism of $E_n$-spaces and determine a principle $GL_1(\mathbb{S})$-bundle over $X$. Then it can be shown in the classical case that there is always a Thom isomor …
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11
votes
1
answer
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Thom Spectra and Hopf-Galois Extensions of Ring Spectra
So I've been fiddling with this for a long time, so apologies to anyone that's already heard me talk about this ad nauseum. I haven't been able to get anywhere with it, and it seemed that as such, it …
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11
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1
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Dedekind spectra
Is there a class of ring spectra that corresponds to and/or extends the class of Dedekind rings from traditional algebra? Is there a notion of "ring of integers" of a ring spectrum? Additionally, is t …
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10
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2
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Geometric Interpretations of Homotopy Theoretical Constructions
In homotopy theory there are lots of nice constructions that seem designed to have some effect on the homotopy of a space, i.e. completing, localizing, and taking various homotopy (co)limits. It seems …
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10
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1
answer
842
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Bar/Cobar Adjunction Between Modules and Comodules
There is a pretty well known, and widely written about, adjunction between augmented algebras and coaugmented coalgebras given by taking the bar construction on algebras and the cobar construction on …
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10
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1
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Why does strong convergence of the EMSS imply that Tot commutes with suspension spectrum?
Given a fiber square of simplicial sets
$$\begin{array}{cc}
& \hspace{-7mm} E \\
&\hspace{-7mm}\downarrow \\
\ast\longrightarrow &\hspace{-7mm} B
\end{array}$$
and a homolo …
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9
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1
answer
289
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Essential maps of spectra which are null when localized at any prime
There are maps of spaces which are not null-homotopic, but when localized at any prime become null. I don't know explicit constructions of any, but an example is given in Section 6 of Chapter 25 of th …
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9
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1
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406
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Shriek push-forward for parameterized spectra
In May and Sigurdsson's Parameterized Homotopy Theory, Proposition 2.2.11, four isomorphisms of functors are given. For a pullback square of base spaces $C=holim(A\overset{f}\to B\overset{j}\leftarrow …
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The relation between the motivic Galois group and the motivic Steenrod algebra
There is a point of view on the Steenrod algebra that goes something like the following: the functor $-\otimes H\mathbb{F}_p\colon Mod_{\mathbb{S}}\to Mod_{H\mathbb{F}_p}$ corresponds to pulling back …
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8
votes
1
answer
438
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Monoidal Model Categories with Suspension Functor
This is basically just me trying to find out what such categories are called, and where they are written about. If I think of some model category of spectra being a "stabilization" of some model cate …
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