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Results tagged with homotopy-theory
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user 9684
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.
21
votes
Integral cohomology (stable) operations
A possibly interesting analogue of the formula $H\mathbb{F}_{2*} H\mathbb{F}_2 = \otimes_{i\ge1} \mathbb{F}_2[\xi_i]$ is $H\mathbb{Z}_{(2)*} H\mathbb{Z}_{(2)} = \bigotimes^\mathbb{L}_{i\ge1} \mathbb{ …
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19
votes
Accepted
Realizing $\mathcal{A}(2)//\mathcal{A}(1)$ by a finite spectrum
The $A(2)$-module structure on $A(2)//A(1)$ does not extend to an $A$-module structure. In particular, there is no spectrum $X$ with $H^*(X; F_2) = A(2)//A(1)$ as an $A(2)$-module.
Additively, $A(2) …
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19
votes
Accepted
Open subspaces of CW complexes
It is not generally true that each open subset of a CW complex admits the structure of a CW complex. Counterexamples were given in
\bib{MR1157891}{article}{
author={Cauty, Robert},
title={Sur …
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17
votes
Differentials in the Adams Spectral Sequence for spheres at the prime p=2
With the aid of machine computations, you can readily determine the Adams differentials up to $t-s=30$ using the multiplicative structure, the relation between Steenrod operations in $\text{Ext}_A$ an …
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15
votes
0
answers
550
views
How well-defined is $\bar\kappa$ in the stable $20$-stem?
The $2$-completed stable $20$-stem $\pi_{20}(S)_2$ is cyclic of order $8$.
Mimura and Toda (1963, Lemma 15.4) mr=157384 show the existence of a class $\bar\kappa_7 \in \pi_{27}(S^7)$ whose stable cla …
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15
votes
What is π_1(BG) for an arbitrary topological group $G$?
The first reference in this general area was:
N. E. Steenrod, "Milgram's classifying
space of a topological group", Topology
7 (1968) 349–368.
Working in the category of compactly generated …
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14
votes
What are some good examples of spectral sequences which degenerate after the first nontrivia...
Some examples with one nonzero family of differentials:
The classical Adams spectral sequence for $j/p$, the connective image-of-J spectrum reduced mod $p$, collapses at $E_3$, by Theorems 4.5 (at $p= …
13
votes
Accepted
Derivations in the Steenrod algebra
The $D$ with $D(xy) = xD(y) + D(x)y$ are the primitives in the Steenrod algebra $A$, which are dual to the indecomposables $\xi_i$ in $A_* = F_2[\xi_i \mid i\ge1]$, so there is one such $D$ in each de …
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11
votes
Accepted
Vietoris-Begle theorem for simplicial sets
(1) As stated, the answer to the question is "no".
Let $A = \Delta[1] \cup_{\partial\Delta[1]} \Delta[1]$ be the union of two copies of $\Delta[1]$ along their common boundary, let $g \colon A \to \D …
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10
votes
Group of units of a ring spectrum vs of its connective cover
For symmetric ring spectra $R$ there is also a definition of the graded group of units, $GL_1^J(R)$, which retains information about the negative homotopy groups of $R$. See Sagave-Schlichtkrull, "Dia …
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10
votes
What is the relation between the sphere spectrum and supersymmetry?
Like Schreiber does in his post, I would advertise the point of view developed by Sagave and Schlichtkrull in their Adv. Math 2012 paper, and used by us to study topological logarithmic geometry. Each …
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9
votes
Does every map $K(\mathbb{Z}, n) \to K(\mathbb{Z}/m, n + k)$ factor through $K(\mathbb{Z}/m,...
The answer is (also) yes when $m=p$ is an odd prime, by Theoreme 2 in
Cartan, H. Détermination des algèbres $H_*(\pi, n; Z_p)$ et $H^*(\pi, n; Z_p)$, $p$ premier impair.
Séminaire Henri Cartan, Tome 7 …
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8
votes
Homotopy fixed points of complex conjugation on $BU(n)$
I think the answer is yes, after Bousfield-Kan $2$-completion. For $n=1$, $BO(1) \to BU(1)^{hC_2}$ is an equivalence, since $BO(1) \simeq K(\mathbb{Z}/2, 1)$, while $BU(1) \simeq K(\mathbb{Z}(1), 2)$ …
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8
votes
Accepted
What is the topological Hochschild cohomology of $\mathbb{F}_p$?
Let me write $HH^S(B) = THH(B) = B \wedge_{B^e} B$ for topological Hochschild homology, and $HH_S(B) = F_{B^e}(B, B)$ for topological Hochschild cohomology, where $B^e = B \wedge_S B^{op}$. For $B$ c …
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8
votes
Accepted
$BP$-Adams Novikov Spectral Sequence or Homotopy groups of $S/3$
For $3$-primary homotopy of $S$ there is early work by
Nakamura, Osamu
Some differentials in the mod 3 Adams spectral sequence.
Bull. Sci. Engrg. Div. Univ. Ryukyus Math. Natur. Sci. No. 19 (1975), 1– …
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