# nLab Poincaré duality algebra

### Context

#### Noncommutative geometry

noncommutative geometry

(geometry $\leftarrow$ Isbell duality $\to$ algebra)

# Contents

## Idea

Generally, a Poincaré duality dg-algebra is a dg-algebra with structure mimicking Poincaré duality in ordinary cohomology.

On the other hand a Poincaré duality $C^\ast$-algebra is a C*-algebra which represents a space in noncommutative topology for which there is a sensible notion of Poincaré duality in K-theory (operator K-theory/K-homology).

## Definition

### For graded-commutative algebras

###### Definition

The structure of a Poincaré duality algebra in dimension $n$ on a graded-commutative graded associative algebra $A$ is a linear function $\epsilon \colon A_n \to k$ to the ground field such that all the induced bilinear forms

$A_k \otimes A_{n-k} \stackrel{\otimes}{\to} A^n \stackrel{\epsilon}{\to} k$

are non-degenerate.

e.g. (Lambrechst-Stanley 07)

### For $C^\ast$-algebras

For C*-algebras hence in noncommutative topology there is the following notion of Poincaré duality, which is really Poincaré with respect not to ordinary cohomology but K-theory (operator K-theory).

We start with the definition of Poincaré self-duality and then generalize to Poincaré dual pairs.

###### Definition

A separable C*-algebra $A \in$ C*Alg is a Poincaré duality algebra (or PD algebra, for short ) if it is dualizable object when regarded as an object of the KK-theory-category, with dual object its opposite algebra.

The element $\Delta$ in def. is called a fundamental class of $A$.

This appears as (BMRS 07, def. 2.1, following Connes, p. 601) following (Connes).

###### Remark

Explicitly def. says that $A$ is a PD algebra if there exists $\Delta \in KK(A \otimes A^{op}, \mathbb{C})$ and $\Delta^\vee \in KK(\mathbb{C}, A \otimes A^{op})$ such that

$\Delta^\vee \otimes_{A^{op}} \Delta = id_A \in KK(A,A)$

and

$\Delta^\vee \otimes_A \Delta = id_{A^{op}} \in KK(A^{op}, A^{op}) \,.$
###### Definition/Proposition

For $A$ $B$ two Poincaré duality algebras, def. Cohomological quantization of local prequantum boundary field theory, master thesis, August 2013

Last revised on August 19, 2013 at 16:30:08. See the history of this page for a list of all contributions to it.