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Poincaré dualization and Massey products

Aleksandar Milivojevic, Jonas Stelzig, Leopold Zoller

Abstract

We study the rational homotopy theoretic and geometric properties of a construction which extends any cohomologically connected, finite type cdga to one satisfying cohomological Poincaré duality. Using this construction we show that non-trivial quadruple Massey products can pull back trivially under non-zero degree maps of Poincaré duality spaces, unlike the case of triple Massey products as studied by Taylor. We also show that a non-zero degree map between formal rational Poincaré duality spaces need not be formal. Our consideration of Massey products naturally ties in with cyclic $A_\infty$-algebras modelling Poincaré duality spaces.

Poincaré dualization and Massey products

Abstract

We study the rational homotopy theoretic and geometric properties of a construction which extends any cohomologically connected, finite type cdga to one satisfying cohomological Poincaré duality. Using this construction we show that non-trivial quadruple Massey products can pull back trivially under non-zero degree maps of Poincaré duality spaces, unlike the case of triple Massey products as studied by Taylor. We also show that a non-zero degree map between formal rational Poincaré duality spaces need not be formal. Our consideration of Massey products naturally ties in with cyclic -algebras modelling Poincaré duality spaces.
Paper Structure (10 sections, 39 theorems, 76 equations)

This paper contains 10 sections, 39 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Poincaré dualization
  4. A quadruple Massey product pulling back trivially
  5. Construction of the example
  6. Homotopical properties of $B$ and its Poincaré dualization
  7. Poincaré dualization for $A_{\infty}$-algebras
  8. A non-formal non-zero degree map
  9. Geometric interpretation of the dualization
  10. Simple Poincaré duality models

Key Result

Theorem A

For every $n$ and any cohomologically connected, finite type cdga $A$, cohomologically concentrated in degrees $<n$, the square-zero extension by its shifted dual $P_nA:=A\oplus D_nA$ is a new cdga satisfying $n$-dimensional Poincaré duality on its cohomology. This construction has the following pro

Theorems & Definitions (89)

  • Theorem A
  • Theorem B
  • Theorem C
  • Proposition 2.1
  • proof
  • Remark 2.2
  • Remark 2.3
  • Example 2.4
  • Example 2.5
  • Definition 3.1
  • ...and 79 more