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Codimension 2 transfer of signatures in L theory

Yuetong Luo

TL;DR

The paper constructs a codimension-2 transfer in L-theory that mirrors known K-theoretic results and shows how the L-signature transfers along a codimension-2 submanifold N inside M. Central to the construction is a suspension-based framework linking L-theory of ΣR with the algebraic L-theory of R, realized via a functor Θ: M^h(ΣR) → 𝔽_{ℕ,b}(M^h(R)) and an induced isomorphism in L-theory. The transfer map ρ_{M,N} is built from a group homomorphism ρ: ℤΓ → ΣℤΠ and the algebraic splitting map, yielding ρ_{M,N}(σ(f,b)) = σ^{<-1>} (f|_{N'},b|_{N'}) and, for homotopy equivalences, 4(Sgn^L(N) - f_*Sgn^L(N')) = 0 in the appropriate L-group. The work leverages Ranicki’s L-theory, assembly/partial-assembly techniques, and the ball-complex ad theory to provide both simplicial and geometric descriptions supporting the transfer theorem. The appendices supply the detailed constructions of Poincaré pairs and the δ-map for cobordism groups, completing the algebraic framework necessary for the L-theoretic transfer result.

Abstract

The signature of a closed manifold is an important geometric topology. Let $M$ be a closed manifold and $N$ be a codimension 2 submanifold of it. Given certain homotopy conditions, Higson, Xie and Schick proved an invariance theorem in codimension 2 for the $K$-theoretic signature. They asked for the $L$-theoretic counterpart of their result. In this note, we will answer their question and moreover, construct a tranfer map between the symmetric $L$-groups of the fundamental groups of $M$ and $N$, which carries the signature of $M$ to that of $N$ up to a torsion of order at most $4$.

Codimension 2 transfer of signatures in L theory

TL;DR

The paper constructs a codimension-2 transfer in L-theory that mirrors known K-theoretic results and shows how the L-signature transfers along a codimension-2 submanifold N inside M. Central to the construction is a suspension-based framework linking L-theory of ΣR with the algebraic L-theory of R, realized via a functor Θ: M^h(ΣR) → 𝔽_{ℕ,b}(M^h(R)) and an induced isomorphism in L-theory. The transfer map ρ_{M,N} is built from a group homomorphism ρ: ℤΓ → ΣℤΠ and the algebraic splitting map, yielding ρ_{M,N}(σ(f,b)) = σ^{<-1>} (f|_{N'},b|_{N'}) and, for homotopy equivalences, 4(Sgn^L(N) - f_*Sgn^L(N')) = 0 in the appropriate L-group. The work leverages Ranicki’s L-theory, assembly/partial-assembly techniques, and the ball-complex ad theory to provide both simplicial and geometric descriptions supporting the transfer theorem. The appendices supply the detailed constructions of Poincaré pairs and the δ-map for cobordism groups, completing the algebraic framework necessary for the L-theoretic transfer result.

Abstract

The signature of a closed manifold is an important geometric topology. Let be a closed manifold and be a codimension 2 submanifold of it. Given certain homotopy conditions, Higson, Xie and Schick proved an invariance theorem in codimension 2 for the -theoretic signature. They asked for the -theoretic counterpart of their result. In this note, we will answer their question and moreover, construct a tranfer map between the symmetric -groups of the fundamental groups of and , which carries the signature of to that of up to a torsion of order at most .
Paper Structure (15 sections, 50 theorems, 480 equations)

This paper contains 15 sections, 50 theorems, 480 equations.

Table of Contents

  1. Introduction
  2. Transfer map
  3. Locally finite $\mathbb{N}$-graded category at infinity
  4. Construction of the category $\mathbb{F}_{\mathbb{N},b}(\mathbb{A})$
  5. The $L$-theory of the category $\mathbb{F}_{\mathbb{N},b}(\mathbb{A})$
  6. Proof of Theorem \ref{['suspensionL']}
  7. Matrix ring of $\Sigma R$
  8. The functor from $M^h(\Sigma R)$ to $\mathbb{F}_{\mathbb{N},b}(M^h(R))$
  9. Proof of Theorem \ref{['L theory transfer']}
  10. Simiplical descriptions of $L$-theory
  11. Ball complex description of $L$-theory
  12. Proof of Theorem \ref{['L theory transfer']}
  13. Appendix
  14. "Poincare pair" construction
  15. Description of the map $\delta:T^k(\underline{L_1},\underline{L_2}) \longrightarrow T^{k+1}(\underline{L_0},\underline{L_1})$

Key Result

Theorem 1.1

Let $M$ be a closed, connected and oriented manifold of dimension $m$ and $N \subset M$ be a connected submanifold of codimension 2 with trivial normal bundle. Assume that the induced map $\pi_1(N) \longrightarrow \pi_1(M)$ is injective and $\pi_2(N) \longrightarrow \pi_2(M)$ is surjective. Then the In particular, if $f$ is a homotopy equivalence, then we get:

Theorems & Definitions (154)

  • Theorem 1.1: Theorem 1.3 in Kubotatransfer
  • Theorem 1.2
  • Remark 2.2
  • Definition 3.1: suspension ring
  • Theorem 3.2
  • Definition 4.1: Additive category with involution
  • Definition 4.2
  • Definition 4.3
  • Definition 4.4: Locally finite $\mathbb{N}$-graded category
  • Definition 4.5: Locally finite $\mathbb{N}$-graded category at infinity
  • ...and 144 more