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Results tagged with homotopy-theory
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user 16785
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.
7
votes
Accepted
Computing Homotopy Fixed Point Spectral Sequences related to Morava E thories
In Ravenel's paper on the Arf invariant, he shows (p. 439) that there is a composite of maps
$$
\mathrm{Ext}_{BP_*BP}(BP_*,BP_*) \to H^*_c(\mathbb{S}_n,E_*) \to H^*(C_p,E_*/\frak{m}),
$$
under which t …
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10
votes
Does an H-space have at most one delooping?
The real projective spaces $\mathbb{R}P^3 \cong SO(3)$ and $\mathbb{R}P^7$ also give fun examples. Naylor proved that there exist 768 $H$-space structures on $SO(3)$, while Rees shows that there exist …
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5
votes
The cooperations algebras Johnson-Wilson theory and truncated BP-theory
Regarding $E(n)_*E(n)$, see "On the Structure of the Hopf Algebroid $E(n)_*E(n)$" by Keith Johnson. Johnson shows that $$E(n)_*E(n) \otimes \mathbb{Q} \simeq \mathbb{Q}[v_1,\cdots,v_{n-1},v_n^{\pm 1}, …
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9
votes
Multiplicative Structures on Moore Spectra
I think the answer to your question is essentially unknown. As far as I'm aware the best known results are:
$M(p)$ admits an $A_{p-1}$ structure but never an $A_p$-structure. I learnt the following …
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12
votes
Accepted
Detecting the Brown-Comenetz dualizing spectrum
Your question appears to be equivalent to the 'dichotomy conjecture' of Hovey, which I believe is still open.
First, note that any finite spectrum has a type, and all finite spectrum of type $n$ ha …
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9
votes
Accepted
$K_3(\mathbb{Z})$ and $\pi ^S_3$
We have $\pi_3(\mathbb{S}) \cong \mathbb{Z}/24\{ \nu\}$ and $\pi_3K(\mathbb{Z}) \cong \mathbb{Z}/48\{ \lambda \}$. As Achim suggested, the unit map $\mathbb{S} \to K(\mathbb{Z})$ induces on $\pi_3$ th …
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12
votes
1
answer
2k
views
Some calculations with the Adams spectral sequence and the cobar complex
I am trying to 'get my hands dirty', so to speak, with some of the calculations with the Adams spectral sequence in Ravenel's Complex Cobordism book, and I have a few questions (I hope it is OK to ask …
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10
votes
0
answers
487
views
Differentials in the Adams-Novikov spectral sequence and the geometric boundary theorem
$\newcommand\Ext{\mathrm{Ext}}
\newcommand\Z{\mathbb{Z}}
\newcommand\G{\mathbb{G}}$
The reference for this question will be the paper by Henn, Karamanov and Mahowald - "The homotopy of the $K(2)$-loca …
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9
votes
1
answer
467
views
Morava modules and completed $E$-homology
Let $E = E_n$ be the $n$-th Morava $E$-theory and let $\{ M_{I} \}$ be a tower of generalised Moore spectra. Then (see this previous question) there is a Milnor exact sequence
$$0 \to \varprojlim_I { …
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6
votes
Accepted
Basic questions on spectra
Here is a slightly more fleshed out version of the comment above.
First, the claim that the collection $\mathcal{C} = \{ \Sigma^{p,q} U \mid U \in Sm/S, p,q \in \mathbb{Z} \}$ is a collection of com …
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32
votes
Why should I care about topological modular forms?
TMF has been used to solve classical topological problems. For example Bruner, Davis and Mahowald obtained new results regarding nonimmersions of real projective spaces in Euclidean space (http://hopf …
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