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4 votes
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Does a complex-oriented $E_1$ ring spectrum (not assumed to have graded-commutative homotopy groups) receive a map from $MU$?

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 ...
John Rognes's user avatar
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24 votes

What are some toy models for the stable homotopy groups of spheres?

My favorite warmup example to the stable homotopy groups of spheres is the following differential graded algebra. Let $A$ have the underlying ring $$ \Bbb Z[y] \otimes \Lambda[x], $$ a ring with a ...
Tyler Lawson's user avatar
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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 ...
John Palmieri's user avatar
15 votes

What are some toy models for the stable homotopy groups of spheres?

You could say that I've made a living out of looking at the stable module category of a finite group (or rather its slight enlargement, the homotopy category of complexes of injective modules, $\...
Dave Benson's user avatar
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5 votes
Accepted

The $E$-(co)homology of $\mathrm{BGL}(R)^+$ and the algebraic $K$-theory of $R$

Let $R$ be a ring. $BGL(R)^+$ is homotopy equivalent to the $0$ component of $\Omega^\infty K(R)$, and it is stably equivalent to $BGL(R)$. In particular, for a (co)homology theory $E$, understanding $...
Maxime Ramzi's user avatar
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0 votes

Which positive flat stable model structures on (flavors of) spectra have the property that cofibrant operad-algebras forget to cofibrant spectra?

I asked this question over a year ago, when revising my paper (joint with Donald Yau) Smith Ideals of Operadic Algebras in Monoidal Model Categories. In the end, I thought this question was relevant ...
David White's user avatar
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7 votes

Is $KU\otimes S^1_+$ isomorphic to $F(S^1_+,KU)$ as $E_\infty$ rings?

This seems to be an answer, based on discussion with Maxime Ramzi in the comments. The space of $E_\infty$ $KU$-algebra maps from $KU\otimes S^1_+\to F(S^1_+,KU)$ is the same as the space of $E_\infty$...
Neil Strickland's user avatar

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