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Results tagged with homotopy-theory
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user 163893
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.
3
votes
0
answers
228
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A modern way to say "$G$ is compact and connected with torsion-free fundamental group"?
Let me start by saying that these ideas are not due to me. I overheard them in a seminar I attended recently (see Footnote).
There are many situations in which one is working with a compact Lie group …
- 2,052
9
votes
0
answers
270
views
A kind of algebraic sphere?
Let $Sch$ be the 1-category of schemes. Is there a cosimplicial scheme $D^\bullet$ and a sequence of schemes $S_1, S_2, ...$ such that the geometric realization of the simplicial set $Hom_{Sch}(D^\bul …
- 2,052
4
votes
1
answer
188
views
Are Landweber exact spectra determined by their coefficient ring?
Let $E$ be a Landweber exact ring spectrum. That is, we have a map of homotopy ring spectra $MU\rightarrow E$ and an isomorphism of homology theories $E_*X\simeq MU_*X\otimes_{MU_*}E_*$. Is the homoto …
- 2,052
7
votes
Does every complex orientable $E_\infty$-ring admit an $E_\infty$ complex orientation?
The answer is no. I guess I owe Fernando Muro 10 dollars.
Let $\mathbb{S}\rightarrow\Sigma^{-2}\mathbb{CP}^\infty$ be the inclusion of the bottom cell, and let $f:F\rightarrow\mathbb{S}$ be the fiber. …
- 2,052
12
votes
1
answer
357
views
Can the Bousfield class of projective space be computed directly?
Recall that the Bousfield class of a spectrum $E$, written $\langle E\rangle$, is the class of spectra $X$ such that $X\wedge E$ is not contractible. For example the Bousfield class of any of the sphe …
- 2,052
10
votes
1
answer
391
views
Does every complex orientable $E_\infty$-ring admit an $E_\infty$ complex orientation?
A ring spectrum $E$ is complex oriented if it is equipped with a ring map $MU\rightarrow E$. It is complex orientable if such a ring map exists.
An $E_\infty$-ring $E$ is $E_\infty$-complex oriented i …
- 2,052
16
votes
3
answers
775
views
"Phantom" non-equivalences of spectra?
I would like an example of the following situation, or a proof that no such example exists.
$\textbf{Situation}$: Two connective (EDIT: I'm fine with dropping this condition) spectra $X$ and $Y$ such …
- 2,052
4
votes
"Phantom" non-equivalences of spectra?
Here's a connective example. It is also an example of Maxime's variant question in the comments (regarding $\tau_{\leq m}$ truncations). And thanks to Maxime for looking this argument over before I po …
- 2,052
12
votes
1
answer
281
views
Is every complex oriented ring spectrum with additive FGL an Eilenberg-Maclane spectrum?
Suppose $E$ is a complex-oriented ring spectrum whose formal group law is isomorphic to the additive one. As the title suggests, we might as well change the complex orientation so that the formal grou …
- 2,052
6
votes
Accepted
Chern classes of a mapping torus vector bundle in terms of the construction data
In the case that $E$ is trivial, there is a "universal" example of the construction you describe, which is the vector bundle on $S^1\times U(n)$ formed the "canonical" automorphism (each point acts o …
- 2,052
4
votes
Accepted
Is every complex oriented ring spectrum with additive FGL an Eilenberg-Maclane spectrum?
This is an answer to the question in the title, which is what I had meant to ask: is an $E$ as in the question body an $H\mathbb{Z}$-module? (the last sentence of the question body is stronger and lik …
- 2,052
26
votes
1
answer
828
views
Are complex-oriented ring spectra determined by their formal group law?
To every complex-oriented ring spectrum $E$ there is associated a formal group law, which is a power series $F_E(x,y)\in E_*[[x,y]]$.
Suppose $E$ and $F$ are two complex-oriented ring spectra and supp …
- 2,052
12
votes
Accepted
Does the spectrum of Morava E-theory depend only on height?
Here's an argument that Eric Peterson and I came up with showing that the homotopy type of Morava $E$-theory only depends on the choice of perfect char $p$ field $k$ and the height $n<\infty$.
$\textb …
- 2,052
3
votes
Accepted
What is the pointed Borel construction of the $0$-sphere?
Let's apply your definition (which I think has typos on the RHS - the two "+" subscripts on the EG should not be there I think). Let's model everything as topological spaces and do the calculation the …
- 2,052