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Connected power operations and simplicial Poincaré duality

Federico Cantero-Morán, Aníbal Medina-Mardones

Abstract

We introduce a structure termed ``connected cyclic diagonal'' on a chain complex, which induces stable power operations in its cohomology with the property that negative power operations consistently vanish. This chain level structure is useful to represent power operations for spectra, whose cohomology lacks a cup product. Using a Poincaré duality algebra structure on the integral chains of the standard augmented simplex, we provide an effective method for the construction of cyclic diagonals on a broad class of augmented simplicial objects.

Connected power operations and simplicial Poincaré duality

Abstract

We introduce a structure termed ``connected cyclic diagonal'' on a chain complex, which induces stable power operations in its cohomology with the property that negative power operations consistently vanish. This chain level structure is useful to represent power operations for spectra, whose cohomology lacks a cup product. Using a Poincaré duality algebra structure on the integral chains of the standard augmented simplex, we provide an effective method for the construction of cyclic diagonals on a broad class of augmented simplicial objects.
Paper Structure (43 sections, 26 theorems, 210 equations, 3 figures)

This paper contains 43 sections, 26 theorems, 210 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Steenrod power operations
  3. Unstable diagonals
  4. Stable diagonals
  5. Connected diagonals
  6. Connected diagonals for simplicial objects
  7. Other connected diagonals
  8. Poincaré duality on the augmented simplex
  9. Outline
  10. Preliminaries
  11. Chain complexes and sign conventions
  12. Tensor products of chain maps with the right suspension
  13. Tensor products of chain maps with the left suspension
  14. Categories
  15. Augmented simplicial objects
  16. ...and 28 more sections

Key Result

Theorem 1.2

If $r$ is an odd prime, each $r$-cyclic straightening with duality yields a natural connected $r$-diagonal on the chain complex of an augmented simplicial object in $\mathcal{C}$.

Figures (3)

  • Figure 1: A depiction of $W^{\mathrm{st}}_{*}(3) \otimes C_{*}$. A stable diagonal comes from an unstable diagonal if it vanishes on the part coloured in green.
  • Figure 2: If a stable diagonal for $r=3$ vanishes on the part coloured in green, then $\mathop{\mathrm{P}}\nolimits^i$ vanishes for $i<0$.
  • Figure 4: The $r$-cyclic assemblage map with duality for $\partial \triangle^2$.

Theorems & Definitions (101)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Definition 3.3
  • ...and 91 more