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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.
5
votes
Reference request: Goodwillie tower of the identity
Jean-Louis Loday told me about the extended action by $\Sigma_{j+1}$ in the fall of 1992, after an Oberwolfach talk I gave about the rank filtration of algebraic $K$-theory, where the $\Sigma_j$-repre …
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7
votes
Accepted
Homotopy in Teichmüller space definition: to be or not to be? That is the question
If $H$ is a homotopy from $f$ to $g$, with $f$ and $g$ invertible, then $f^{-1} H g^{-1}$ is a homotopy from $g^{-1}$ to $f^{-1}$.
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7
votes
Accepted
Finite complexes which are not Thom spectra
The proposed argument for why $Q = S \cup_2 e^1 \cup_\eta e^3$ is not a Thom spectrum seems to use that the Thom isomorphism commutes with the Steenrod operations, which is often false. The deviation …
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4
votes
Homology of braid groups and loop spaces
Looping the fiber sequence $S^1 \to S^3 \to S^2$ gives $\Omega^2 S^2 \simeq \mathbb{Z} \times \Omega^2 S^3$. This is the group completion $\mathbb{Z} \times BB_\infty^+$ of $\coprod_{n\ge0} BB_n$, so …
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1
vote
Reference for choosing a path lifting function?
(Not an answer, but long for a comment.) Spanier's "Algebraic Topology", Section 2.7, gives Hurewicz' proof of the theorem that a local (Hurewicz) fibration with respect to a numerable open cover of …
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6
votes
Can any suspension spectrum be realized as Waldhausen K-theory?
(1) Given a simplicial monoid $G$ let $R^0(*, G)$ be the Waldhausen category of pointed finite free $G$-simplicial sets weakly equivalent to $(\coprod^k G)_+$, for varying $k\ge0$. This is the special …
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4
votes
Accepted
When does a map in the stable homotopy group gets killed when smashed with cone of itself?
The answer to your third question is "yes".
A little more generally, if $Sq^{n+1}$ acts nontrivially in the mod $2$ cohomology of $C$, then $Sq^{(k+1)(n+1)}$ acts nontrivially in the mod $2$ cohomolo …
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6
votes
Accepted
Map between homotopy groups of O, related to J-homomorphism and K-theory of Z
The composition is trivial. Composition with $\eta$ acts trivially on the source and nontrivially on the target, for positive $s$, which implies the claim. This argument does not apply for $s=0$, beca …
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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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3
votes
Numerator in the zeta values at negative odd integers
(Too long for a comment:)
As A.S. commented, the absolute value of $\zeta(1-2k) = - B_{2k}/2k$ for $k\ge1$ is realized as twice the order of $K_{4k-2}(Z)$ divided by the order of $K_{4k-1}(Z)$, so for …
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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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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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2
votes
Accepted
Does a filtered A_N algebra give rise to a multiplicative spectral sequence?
For the $d_r$-differentials to be derivations, i.e., to satisfy the Leibniz rule $d_r(x \cdot y) = d_r(x) \cdot y \pm x \cdot d_r(y)$ with $x \cdot y = m_2(x \otimes y)$, it is enough to have a filter …
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6
votes
Non-zero homotopy/homology in diffeomorphism groups
The diffeomorphism groups $\text{Diff}(M)$ are sensitive to stabilization, say replacing $M$ by $M \times [0,1]$, so the direct contribution of the homotopy type of $M$ to $\text{Diff}(M)$ can be obsc …
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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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