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Bounds on Cheeger–Gromov invariants and simplicial complexity of triangulated manifolds

  • Geunho Lim ORCID logo EMAIL logo and Shmuel Weinberger ORCID logo
Published/Copyright: February 16, 2024

Abstract

We show the existence of linear bounds on Wall 𝜌-invariants of PL manifolds, employing a new combinatorial concept of 𝐺-colored polyhedra. As an application, we show how the number of h-cobordism classes of manifolds simple homotopy equivalent to a lens space with 𝑉 simplices and the fundamental group of Zn grows in 𝑉. Furthermore, we count the number of homotopy lens spaces with bounded geometry in 𝑉. Similarly, we give new linear bounds on Cheeger–Gromov 𝜌-invariants of PL manifolds endowed with a faithful representation also. A key idea is to construct a cobordism with a linear complexity whose boundary is π1-injectively embedded, using relative hyperbolization. As an application, we study the complexity theory of high-dimensional lens spaces. Lastly, we show the density of 𝜌-invariants over manifolds homotopy equivalent to a given manifold for certain fundamental groups. This implies that the structure set is not finitely generated.

Award Identifier / Grant number: 2019R1A3B2067839

Award Identifier / Grant number: DMS-2105451

Funding statement: G. Lim was partially supported by the National Research Foundation of Korea grant 2019R1A3B2067839. S. Weinberger was partially supported by the National Science Foundation grant DMS-2105451.

Acknowledgements

We would like to thank Jae Choon Cha and Fedya Manin for very stimulating conversations on a number of topics around the contents of this paper.

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Received: 2023-05-08
Revised: 2024-01-09
Published Online: 2024-02-16
Published in Print: 2024-03-01

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