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$L$-theory Characteristic Classes

Runjie Hu

TL;DR

This paper resolves the long-standing question of whether the local $L$-theory data encapsulated by Levitt–Ranicki orientations for TOP bundles and spherical fibrations can be identified with the classical $2$-local characteristic classes of Brumfiel–Morgan and Morgan–Sullivan, together with Sullivan’s odd-prime $KO$ orientation. Its central achievement is to construct geometric homotopy equivalences that decompose the $2$-local connective $L$-spectra into products of Eilenberg–MacLane spectra and relate odd-prime local data to real $K$-theory, thereby proving the equivalence of the two viewpoints. The work also re-proves known local structures of the $L$-spectra, answers longstanding questions (including Brumfiel’s and Sullivan’s), and clarifies the odd-prime versus even-prime lifting problems for spherical fibrations and TOP bundles. Collectively, these results unify surgery-theoretic invariants across chain-level and geometric formulations, and provide a robust framework for translating $L$-theory orientations into classical characteristic classes with concrete computational tools.

Abstract

Although the local information of the $L$-spectra is well understood, the problem of whether this local information can be identified with the geometric data for bundles remains open for decades, which was originally raised in the 1960s and 1970s by Sullivan, Brumfiel, Taylor-Williams and others independently. In this paper, we provide an affirmative answer by proving that Levitt-Ranicki's theory of connective $L$-orientations for $TOP$ bundles and spherical fibrations is equivalent to the $2$-local characteristic classes constructed by Brumfiel-Morgan's, Madsen-Milgram's and Morgan-Sullivan's, as well as Sullivan's odd-prime-local real $K$-theory orientation. A key step in our proof involves constructing more geometric homotopy equivalences from the $2$-local quadratic, symmetric and normal connective $L$-spectra to products of Eilenberg-Maclane spectra and those from odd-local quadratic and symmetric connective $L$-spectra to the connective real $K$-spectra. This approach reproves the known local structure of $L$-spectra.

$L$-theory Characteristic Classes

TL;DR

This paper resolves the long-standing question of whether the local -theory data encapsulated by Levitt–Ranicki orientations for TOP bundles and spherical fibrations can be identified with the classical -local characteristic classes of Brumfiel–Morgan and Morgan–Sullivan, together with Sullivan’s odd-prime orientation. Its central achievement is to construct geometric homotopy equivalences that decompose the -local connective -spectra into products of Eilenberg–MacLane spectra and relate odd-prime local data to real -theory, thereby proving the equivalence of the two viewpoints. The work also re-proves known local structures of the -spectra, answers longstanding questions (including Brumfiel’s and Sullivan’s), and clarifies the odd-prime versus even-prime lifting problems for spherical fibrations and TOP bundles. Collectively, these results unify surgery-theoretic invariants across chain-level and geometric formulations, and provide a robust framework for translating -theory orientations into classical characteristic classes with concrete computational tools.

Abstract

Although the local information of the -spectra is well understood, the problem of whether this local information can be identified with the geometric data for bundles remains open for decades, which was originally raised in the 1960s and 1970s by Sullivan, Brumfiel, Taylor-Williams and others independently. In this paper, we provide an affirmative answer by proving that Levitt-Ranicki's theory of connective -orientations for bundles and spherical fibrations is equivalent to the -local characteristic classes constructed by Brumfiel-Morgan's, Madsen-Milgram's and Morgan-Sullivan's, as well as Sullivan's odd-prime-local real -theory orientation. A key step in our proof involves constructing more geometric homotopy equivalences from the -local quadratic, symmetric and normal connective -spectra to products of Eilenberg-Maclane spectra and those from odd-local quadratic and symmetric connective -spectra to the connective real -spectra. This approach reproves the known local structure of -spectra.
Paper Structure (23 sections, 140 theorems, 233 equations)

This paper contains 23 sections, 140 theorems, 233 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. lg Groups
  4. lg Spectra
  5. A Priori Invariants by Periods
  6. lg-Manifolds
  7. Cohomological and $K$-Theoretical A Priori Invariants
  8. Applications to Surgery Theory
  9. Lg-theory with Lg coefficient
  10. lg Chains and their bordisms
  11. Product of lg Chains
  12. Product Structure of Lg-Theory with Lg coefficient
  13. Symmetric and Quadratic Case
  14. Normal Case
  15. Lg-theory Characteristic Classes and Bundle Lifting Problem
  16. ...and 8 more sections

Key Result

Theorem 1.3

Ranicki-Levitt's theory of connective $L$-orientations for $TOP$ bundles and spherical fibrations is equivalent to Brumfiel-Morgan's, Madsen-Milgram's and Morgan-Sullivan's $2$-local characteristic classes and Sullivan's odd-prime-local $KO$-orientation.

Theorems & Definitions (195)

  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.6
  • Corollary 1.7
  • Remark 2.1
  • Lemma 2.2
  • proof
  • Corollary 2.3
  • proof
  • Remark 2.4
  • ...and 185 more