[Paper Review] Stable moduli spaces of hermitian forms

Fabian Hebestreit, Wolfgang Steimle|arXiv (Cornell University)|Mar 25, 2021

Homotopy and Cohomology in Algebraic Topology40 references4 citations

TL;DR

This paper establishes that Grothendieck-Witt spaces of Poincaré categories are group completions of moduli spaces of hermitian forms, thereby identifying Karoubi's classical hermitian and quadratic K-groups with the refined Grothendieck-Witt groups from recent work. The proof adapts techniques from Galatius and Randal-Williams on cobordism categories, using the homotopy type of the cobordism category to realize the group completion as a colimit over iterated Q-constructions.

ABSTRACT

We prove that Grothendieck-Witt spaces of Poincaré categories are, in many cases, group completions of certain moduli spaces of hermitian forms. This, in particular, identifies Karoubi's classical hermitian and quadratic K-groups with the genuine Grothendieck-Witt groups from our joint work with Calmès, Dotto, Harpaz, Land, Moi, Nardin and Nikolaus, and thereby completes our solution of several conjectures in hermitian K-theory. The method of proof is abstracted from work of Galatius and Randal-Williams on cobordism categories of manifolds using the identification of the Grothendieck-Witt space of a Poincaré category as the homotopy type of the associated cobordism category.

Motivation & Objective

• To identify Karoubi's classical hermitian and quadratic K-groups with the modern Grothendieck-Witt groups from recent higher algebraic K-theory.
• To establish that the Grothendieck-Witt space of a Poincaré category is the group completion of the moduli space of hermitian forms.
• To complete the solution of long-standing conjectures in hermitian K-theory by providing a homotopical identification of classical and modern K-theory invariants.
• To extend the framework of parametrized algebraic surgery and surgery complexes to Poincaré categories, enabling the construction of group completions via iterated Q-constructions.

Proposed method

• Adapts the method of Galatius and Randal-Williams on cobordism categories, identifying the Grothendieck-Witt space of a Poincaré category with the homotopy type of its associated cobordism category.
• Uses the identification of the $Χ$-construction on the $Χ$-category of Poincaré categories as a model for the group completion of the moduli space of hermitian forms.
• Applies the weight theorem for $ι$-spaces (by Harpaz) to relate the homotopy type of the cobordism category to the $Χ$-construction on the heart of the category.
• Employs the $Χ$-construction and iterated Q-constructions to realize the group completion as a colimit of iterated bar constructions on the $Χ$-group of the heart of the category.
• Uses the fact that the $Χ$-construction on the $Χ$-category of a Poincaré category is equivalent to the edgewise subdivision of the S-construction, enabling homotopical control.
• Applies a general criterion for natural transformations between $ε_{\infty}$-group-valued functors to be equivalences when they are so on the heart of the category, via bicartesian squares and colimit arguments.

Experimental results

Research questions

• RQ1 Is the Grothendieck-Witt space of a Poincaré category the group completion of the moduli space of hermitian forms?
• RQ2 Can Karoubi's classical hermitian and quadratic K-groups be identified with the refined Grothendieck-Witt groups from modern higher algebraic K-theory?
• RQ3 Does the homotopy type of the cobordism category of a Poincaré category model the group completion of the moduli space of hermitian forms?
• RQ4 Can the $Χ$-construction on the $Χ$-category of a Poincaré category be used to realize the group completion via iterated Q-constructions?
• RQ5 Under what conditions does the natural transformation between $Χ$-group-valued functors on Poincaré categories induce an equivalence on the Grothendieck-Witt space?

Key findings

• The Grothendieck-Witt space of a Poincaré category is the group completion of the moduli space of hermitian forms, establishing a homotopical identification of classical and modern K-theory invariants.
• Karoubi's classical hermitian and quadratic K-groups are identified with the refined Grothendieck-Witt groups from the authors' joint work, completing the solution of several conjectures in hermitian K-theory.
• The homotopy type of the cobordism category of a Poincaré category realizes the group completion of the moduli space of hermitian forms, generalizing results from manifolds to algebraic settings.
• The $Χ$-construction on the $Χ$-category of a Poincaré category is equivalent to the edgewise subdivision of the S-construction, enabling homotopical control via bar constructions.
• The colimit of iterated $Χ$-constructions on the heart of a Poincaré category vanishes, which implies that the natural transformation between $Χ$-group-valued functors is an equivalence.
• For a 0-dimensional Poincaré category with ordinary heart and abelian values in spectra, the map from the Grothendieck-Witt space of the heart to the Grothendieck-Witt space of the category is an equivalence.