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Results tagged with homotopy-theory
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user 4194
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.
9
votes
Accepted
The Bousfield Class of the Infinite Wedge of Telescopes of Finite Spectra
The spectrum $H\mathbf{F}_p$ is an acyclic for both $\bigvee \langle T(n) \rangle$ and $\bigvee \langle K(n) \rangle$. Therefore neither of these wedges equals $\langle S \rangle$.
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11
votes
1
answer
292
views
W H Lin's thesis and Hopf subalgebras of the Steenrod algebra
If $B$ is a subalgebra of $A$, you can ask whether the $B$-module structure on $B$ can be extended to give an $A$-module structure on $B$.
W H Lin, in his 1973 PhD thesis at Northwestern, showed that …
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11
votes
Accepted
W H Lin's thesis and Hopf subalgebras of the Steenrod algebra
Using @CarloBeenakker's answer, our librarian found an electronic version, produced from the microfilm copy of the original: https://search.proquest.com/docview/302701183 (full text may require access …
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11
votes
What are some good examples of spectral sequences which degenerate after the first nontrivia...
The Serre spectral sequence for the path-loop fibration for an $n$-sphere is a positive answer to question 1, a negative answer to question 2. More generally, a fibration in which either the base or t …
8
votes
Simplicial model of Hopf map?
There is a small simplicial set description of the Hopf map in Clemens Berger's thesis, Exemple 1.19, pp. 45-47.
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8
votes
What are some toy models for the stable homotopy groups of spheres?
As Dave Benson says, the Noetherian condition simplifies a lot of things. The derived category of a commutative ring satisfies many of the properties of the stable homotopy category. The derived categ …
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8
votes
Algebraic topology and homotopy theory with simplicial sets instead of topological spaces
It certainly used to be the case that if you went to an algebraic topology research talk, when the speaker said, "Let $X$ be a space," then there was about a 50% chance that they really meant "Let $X$ …
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4
votes
Derivations in the Steenrod algebra
I have a guess for question 1. Fix $n \geq 0$ and let $E(n)$ be the Hopf subalgebra dual to $\mathbb{F}_2 [\xi_{n+1}, \xi_{n+2}, \dots] / (\xi_i^{2^{n+1}})$. Every $x\in E(n)$ satisfies $x^2=0$, and m …
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5
votes
Is there an additive model of the stable homotopy category?
In the category $Ho(C)$, say that every map is a fibration and a cofibration, and define the weak equivalences to be the isomorphisms. This makes $Ho(C)$ into a model category; its homotopy category i …
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15
votes
Accepted
Is homology finitely generated as an algebra?
Another counterexample: let $A$ be the algebra $\mathbb{Q}[y,z]/(y^2) \otimes \bigwedge(x)$ with $x$ in degree 1, $y$ and $z$ in degree 2. Put a differential on this by $z \mapsto xy$. This is a commu …
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