Bounds on Cheeger-Gromov invariants and simplicial complexity of triangulated manifolds
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2023
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| _version_ | 1866917570347532288 |
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| author | Lim, Geunho Weinberger, Shmuel |
| author_facet | Lim, Geunho Weinberger, Shmuel |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_08870 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Bounds on Cheeger-Gromov invariants and simplicial complexity of triangulated manifolds Lim, Geunho Weinberger, Shmuel Geometric Topology Algebraic Topology 57N16, 57N70, 57Q20, 55U10, 55U15 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. |
| title | Bounds on Cheeger-Gromov invariants and simplicial complexity of triangulated manifolds |
| topic | Geometric Topology Algebraic Topology 57N16, 57N70, 57Q20, 55U10, 55U15 |
| url | https://arxiv.org/abs/2301.08870 |