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Results tagged with at.algebraic-topology
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user 102390
Homotopy theory, homological algebra, algebraic treatments of manifolds.
2
votes
Looking for examples of maps $\Omega^lS^{n+l}\to\Omega^kS^{m+k}$ with $l>k$
Localize at a prime $p$. Then (this is from my comment above) there is a fiber sequence
$$J_{p^k−1}S^{2n} \to \Omega S^{2n+1} \to \Omega S^{2npk+1},$$
coming from the combinatorial James-Hopf map (see …
- 5,520
6
votes
0
answers
184
views
The chromatic splitting conjecture and functoriality
Let $M$ be a finite spectrum, so that $L_{K(n)} M = M \wedge L_{K(n)} S$. Recall that (a weak version of) the chromatic splitting conjecture states that the chromatic attaching map $L_{n-1} M \to L_{n …
- 5,520
2
votes
To compare the total, base and fiber spaces of two fiber bundles
Let $f$ be the map between the two Serre fibrations. The question then asks: if $f$ is a rational homotopy equivalence on the base and total spaces, then is it a rational homotopy equivalence on the f …
- 5,520
9
votes
Accepted
Lecture notes by Mahowald and Unell
I scanned the notes (apologies for the delay). Thanks a lot to Peter May for lending me the notes and for letting me scan them! Here's the link: http://www.mit.edu/~sanathd/mahowald-unell-bott.pdf.
- 5,520
11
votes
1
answer
408
views
Finite complexes which are not Thom spectra
I'll be working in the stable world. It's an easy observation that any 2-cell complex (over the sphere) with bottom cell in dimension zero is a Thom spectrum: any such complex is the cofiber of some e …
- 5,520
6
votes
Accepted
Is the mod-2 Moore spectrum a retract of a shift of its tensor square?
The mod $2$ cohomology of $S^0/2 \wedge S^0/2$ is a $\mathbf{F}_2$-vector space on generators in degrees 0, 1, 1, and 2. The classes in degrees 0 and 2 are connected by a nontrivial $\mathrm{Sq}^2$, s …
- 5,520
9
votes
3
answers
734
views
Lecture notes by Mahowald and Unell
I'm trying to find lecture notes of Mahowald and Unell, titled "Lectures on Bott periodicity in stable and unstable homotopy at the prime 2". Does anyone happen to know if a copy exists online (and if …
- 5,520
2
votes
Crafting Suspension Spectra
For large $i_0,\cdots,i_n$ we can realize $BP_\ast/(v_0^{i_0},\cdots,v_n^{i_n})$ as the $BP$-homology of a finite spectrum $S/(v_0^{i_0},\cdots,v_n^{i_n})$. This follows from Devinatz-Hopkins-Smith. S …
- 5,520
8
votes
Accepted
The connective $k$-theory cohomology of Eilenberg-MacLane spectra
Charles Rezk already answered this in the comments; I'll just expand on what he wrote. This paper discusses what's now known as Mahowald-Rezk duality; this is a version of Anderson duality that takes …
- 5,520
17
votes
Accepted
Homology of spectra vs homology of infinite loop spaces
$\newcommand{\H}{\mathrm{H}} \newcommand{\Z}{\mathbf{Z}}$Let $X$ be a space. Then the $E$-(co)homology of $X$ is the same as the $E$-(co)homology of its suspension spectrum, i.e., $E_\ast(X) \cong E_\ …
- 5,520
6
votes
1
answer
491
views
Unstable Greek letter elements
A theorem of Hopkins and Mahowald states that the Thom spectrum of the map $\Omega^2 S^3 \to B\mathrm{GL}_1(\mathbb{S}_{(p)})$ classifying the element $p$ is exactly $\mathrm{H}\mathbf{F}_p$. Let $T(1 …
- 5,520
2
votes
Accepted
Mapping a loop space to quaternionic projective space
[I should've posted this answer a long time ago, given that it was answered in the comments.] In order to extend the map $S^4\to \mathbf{H}P^\infty$ to a map from $J_2(S^4)$, the composite of $[\iota_ …
- 5,520
11
votes
0
answers
432
views
$E_\infty\mathrm{Spaces}(\mathbf{Z}/p\mathbf{Z},GL_1(E_n))$ and Eilenberg-Maclane spaces
$\newcommand{\Z}{\mathbf{Z}}$Let $p$ be a prime. In his answer here, Jacob Lurie conjectured that $E_\infty\mathrm{Spaces}(\mathbf{Z}/p\mathbf{Z},GL_1(E_n))\simeq K(\Z/p\Z,n)$ where $E_n$ denotes the …
- 5,520
16
votes
Accepted
The structure of complex cobordism cohomology of the Eilenberg-Maclane spectrum
One can prove that $\mathrm{Map}(H\mathbf{F}_p,MU)$ is contractible. We know that $H\mathbf{F}_p$ is dissonant (Theorem 4.7 of Ravenel's "Localization with Respect to Certain Periodic Homology Theorie …
- 5,520
2
votes
Accepted
Homology of a loop-suspension space and action of $\mathcal{D}_1$-operad
$\newcommand{\E}{\mathbf{E}} \newcommand{\co}{\mathcal{O}} \newcommand{\free}{\mathrm{Free}} \newcommand{\H}{\mathrm{H}}$Here's one way of seeing the Bott-Samelson theorem. The James splitting gives a …
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