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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.
4
votes
Identifying $d_1$ in the Atiyah-Hirzebruch-Serre spectral sequence
This is Theorem 4.8 in Chapter XIII of
Whitehead, George W.
Elements of homotopy theory.
Graduate Texts in Mathematics, 61.
Springer-Verlag, New York-Berlin, 1978.
xxi+744 pp. ISBN: 0-387-90336-4
an …
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6
votes
Accepted
Loop-space functor on cohomology
If you are willing to read a little French, look at page 434 of Serre's "Homologie singuliere des espaces fibres" (1951).
For an exercise in English, with a hint, try Exercise 2 on page 155 in Mosher …
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5
votes
Group completion of a monoid (braid groups)
See Proposition 1 in
McDuff, D.; Segal, G.
Homology fibrations and the "group-completion'' theorem.
Invent. Math. 31 (1975/76), no. 3, 279–284.
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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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5
votes
Accepted
Does a complex-oriented $E_1$ ring spectrum (not assumed to have graded-commutative homotopy...
The initial example of such an $E$ is the Thom spectrum $M\xi$ associated to the $E_1$-map $\Omega \Sigma BU(1) \to BU$, studied by Baker and Richter in "Quasisymmetric functions from a topological po …
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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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7
votes
The complex $K$-theory of the Thom spectrum $MU$
You can learn about this, and more, in Part II of Adams' "Stable Homotopy and Generalised Homology" (1974), especially section 4, in the special case $E = KU$.
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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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6
votes
Classifying the endofunctors of the category $\Delta$ of finite linear orders
See Edgewise subdivision and simple maps by Knut Berg (supervised by me), Generalized edgewise subdivisions by Katerina Velcheva (supervised by Clark Barwick) and the earlier MathOverflow question Wha …
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3
votes
Defining homotopy via endofunctors of a simplicial category
There is a problem with base points in your claim. If $X$ is discrete, then ${\rm sing} X_\bullet$ and ${\rm sing} X_\bullet \circ [+1]$ are the same, so every map from ${\rm sing} F_\bullet$ factors …
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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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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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4
votes
What is $TP(\mathbb{Z}_p)$?
The calculation of $\pi_* TP(\mathbb{F}_p) = \pi_* THH(\mathbb{F}_p)^{tS^1} = \pi_* \widehat{\mathbb{H}}(S^1, THH(\mathbb{F}_p))$ (the notation has changed over the years) was first published by Hesse …
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7
votes
Accepted
Generalization of Hopf invariant
The $E$-based Adams spectral sequence is the homotopy spectral sequence associated to the tower of spectra $\dots \to Y_2 \to Y_1 \to Y_0 = S$ with $Y_{s+1} \to Y_s \to E \wedge Y_s$ a homotopy fiber …
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