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Torsion Invariants

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Introduction to ℓ²-invariants

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2247))

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Abstract

We explain how ℓ 2-torsion arises as the ℓ 2-counterpart to Reidemeister torsion and give an intuitive picture on how to think about chain complexes in this context. We extract some explicit computations of ℓ 2-torsion for spaces and groups from the literature and we present how twisting the ℓ 2-chain complex with a character leads to the definition of ℓ 2-Alexander torsion. As we will see, the degree of the ℓ 2-Alexander torsion function recovers the Thurston norm for 3-manifolds. Next we give full details how ℓ 2-torsion relates to torsion in homology via regulators. As a consequence, we show how the torsion approximation conjecture actually splits up into three different problems, each of it intriguing and wide open. Specializing to an arithmetic setting, we present the Bergeron Venkatesh conjecture on torsion in twisted homology. We conclude the text with an account on profinite rigidity, (non-)profiniteness of ℓ 2-Betti numbers and a proof that the torsion approximation conjecture implies profiniteness of volume for 3-manifolds.

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  1. Institute for Algebra and Geometry, Karlsruhe Institute of Technology, Karlsruhe, Germany

    Holger Kammeyer

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Kammeyer, H. (2019). Torsion Invariants. In: Introduction to ℓ²-invariants. Lecture Notes in Mathematics, vol 2247. Springer, Cham. https://doi.org/10.1007/978-3-030-28297-4_6

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