Schreiber Rational parameterized stable homotopy theory

An article that we are writing:

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• Vincent Braunack-Mayer, Urs Schreiber

  Rational parameterized stable homotopy theory

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Abstract We combine Quillen-Sullivan’s classical rational homotopy theory with Schwede-Shipley’s stable homotopy theory of HR-module spectra to prove that rational parameterized spectra over a rational parameter space are equivalently modeled by the homotopy theory of dg(co-)modules over the Quillen-Sullivan dg(co-)algebras that models the parameter spaces. This implies also that rational parameterized spectra form a full subcategory of the slice homotopy theory of unbounded $L_\infty$-algebras, thereby extending Hinich‘s embedding of rational homotopy theory in $L_\infty$-algebras from non-negative to undounded degrees. Minimal models of augmented Sulllivan algebras regarded just as minimal dg-modules over the base algebra have been studied before by A. Roig, and our result shows that these are models for the fiberwise suspension spectra of the corresponding rational fibration.

An interesting example is the fiberwise suspension spectrum over the 3-sphere of the homotopy quotient of the 4-sphere $S^4 \simeq S(\mathbb{R} \oplus \mathbb{H})$ by the circle action: this turns out to contain a copy of twisted K-theory, rationally: $\Sigma^\infty_{S^3} (S^4/S^1) \underset{\mathbb{Q}}{\simeq} ku/BU(1) \oplus \cdots$.

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References

• thesis Schlegel