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

Geunho Lim, Shmuel Weinberger

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 application, we show that how the number of h-cobordism classes of manifolds simple homotopy equivalent to a lens space with $V$ simplices and the fundamental group of $\mathbb{Z}_n$ 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 $π_1$-injectively embedded, using relative hyperbolization. As 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.

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

Abstract

We show the existence of linear bounds on Wall -invariants of PL manifolds, employing a new combinatorial concept of -colored polyhedra. As application, we show that how the number of h-cobordism classes of manifolds simple homotopy equivalent to a lens space with simplices and the fundamental group of grows in . Furthermore we count the number of homotopy lens spaces with bounded geometry in . Similarly, we give new linear bounds on Cheeger-Gromov -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 -injectively embedded, using relative hyperbolization. As application, we study the complexity theory of high-dimensional lens spaces. Lastly we show the density of -invariants over manifolds homotopy equivalent to a given manifold for certain fundamental groups. This implies that the structure set is not finitely generated.
Paper Structure (6 sections, 10 theorems, 19 equations, 1 figure)

This paper contains 6 sections, 10 theorems, 19 equations, 1 figure.

Table of Contents

  1. Introduction and main results
  2. Wall $\rho$-invariant and Cheeger-Gromov $L^2$ $\rho$-invariant
  3. $G$-colored polyhedra and their coverings
  4. Quantitative relative hyperbolization
  5. Complexity and homotopy structures of homotopy lens spaces
  6. Density of Cheeger-Gromov $L^2$ $\rho$-invariants over structure sets

Key Result

Theorem 1.1

If $M$ is an odd dimensional PL manifold and one is given a homomorphism $\alpha:\pi_1(M) \longrightarrow G$ for a finite group $G$, then $\lvert \rho_{g}(M) \rvert \leq C(d) \cdot \lvert G \rvert \cdot \Delta (M)$, for any nontrivial element $g$ in $G$.

Figures (1)

  • Figure 1: The hyperbolization process is to fiber-wisely replace an abstract simplex in a barycentric subdivision of a simplicial complex by a hyperbolized simplex. This picture describes $H(\sigma)=X^2 \triangle \sigma =X^2 \widetilde{\triangle} \sigma'$ where $\sigma$ is an abstract $2$-simplex. The hyperbolic $2$-simplex is colored orange, while the barycentric subdivision of $\sigma$ is colored gray.

Theorems & Definitions (47)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • ...and 37 more