# nLab Poincaré duality algebra

### Context

#### Noncommutative geometry

noncommutative geometry

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

integration

# 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

###### 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. 2 is called a fundamental class of $A$.

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

###### Remark

Explicitly def. 2 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. \ref{PDAlgebra}, and for $f \colon A \to B$ a homomorphism between them, regarded as a morphism $f^\ast \colon B \to A$ in KK-theory, the correspondung dual morphism $f! \colon A \to B$ is the one such that postcomposition in $KK$ with this corresponds to the Umkehr map/push forward in generalized cohomology? in KK-theory.

For more on this see below at Properties – K-Orientation and Umkehr mpas.

###### Remark

For $C^\ast$-algebras which are groupoid convolution algebras $C^\ast(\mathcal{G})$ the opposite algebra is Morita equivlant (since a groupoid $\mathcal{G}$ is equivalent to its opposite groupoid $\mathcal{G}^{op}$, the equivalence being induced by the functor which sends a morphism to its inverse). But given a circle 2-bundle $\chi \colon \mathcal{G} \to \mathbf{B}^2 U(1)$ the corresponding twisted groupoid convolution algebra is such that passing to the opposite corresponds to passing to the inverse twist $-\chi$.

Therefore it makes sense to consider more generally

###### Definition

For $A$ a C*-algebra a Poincaré dual for $A$ is a dual object $B \in C^\ast Alg \to KK$ in KK-theory.

Below in the Proposition-Section is discussed how under Poincaré-duality the twist changes.

## Properties

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

#### Duals and twists

###### Proposition

Let $X$ be a closed manifold with spin^c-structure. Then there is a Poincaré duality isomorphism

$K^\bullet(X) \simeq K_\bullet(X) \,.$

For instance (Connes, chapter 2.7, prop. 5).

(…) The relaton between Poincaré duality on algebras of functions and spin^c-structure is discussed in (Connes, around p. 603). (…)

Notice that the obstruction to spin^c structure is the third integral Stiefel-Whitney class $W_3 \colon B SO \to B^2 U(1)$. If this does not vanish on a manifold, then a Poincaré dual/dual object in KK-theory still exists, but is the same manifold equipped with a twist shifted by $W_3(\tau_X)$, where $\tau_X$ denotes the (co)tangent bundle of $X$.

###### Definition/Notation

For $X$ a (compact) manifold and $c \in H^3(X,\mathbb{Z})$ the class of a circle 2-bundle/bundle gerbe $\mathcal{G}$ on $X$, write

$C_c(X) \in C^\ast Alg \to KK$

for the corresponding twisted groupoid convolution algebra, the one whose operator K-theory is the $c$-twisted K-theory of $X$:

$KK_\bullet(\mathbb{C}, C_c(X)) \simeq K_{\bullet + c}(X) \,.$
###### Proposition

Let $X$ be a compact manifold with tangent bundle $\tau_X$ and let $c \in H^3(X,\mathbb{Z})$ be a twist. Then the C*-algebra $C_{c}(X)$ of def. 4 has a dual object in the full subcategory of KK-theory on separable C*-algebras, given by

$(C_c(X))^\vee \simeq C_{\frac{1}{c\otimes W_3(\tau_X)}}(X) \,,$

hence by the same manifold but with twist the inverse of the third integral Stiefel-Whitney class and the original twist.

The same remains true in $G$-equivariant KK-theory, for $G$ a locally compact topological group.

The non-equivariant case is in (Brodzki-Mathai-Rosenberg-Szabo 06, section 7.3) and the generalization to the equivariant case in (Tu 06, theorem 3.1) (where we use remark 2 in order to interpret the opposite twisted convolution algebra up to equivalence as inducing the inverse twist).

#### K-Orientation and Umkehr maps

We discuss Umkehr maps/fiber integration in generalized cohomology in K-theory using Poincaré duality algebras / dual objects in KK-theory.

###### Proposition

Every homomorphism $f \colon A \to B$ between PD $C^\ast$-algebras is K-orientable in KK-theory. The K-orientation is given by the corresponding dual morphism, hence the element $f! \colon B \to A$ given as the composite

$f! \coloneqq \Delta^\vee_A \otimes_{A^{op}} f^{op} \otimes_{B^{op}} \Delta_B \,.$

More generally we have the following.

###### Example

Let $i \colon Q \to X$ be a map of compact manifolds and let $\chi \colon X \to B^2 U(1)$ modulate a circle 2-bundle regarded as a twist for K-theory. Then forming twisted groupoid convolution algebras yields a KK-theory morphism of the form

$C_{i^\ast \chi}(Q) \stackrel{i^\ast}{\longleftarrow} C_{\chi}(X) \,,$

with notation as in def. 4. By prop. 3 the dual morphism is of the form

$C_{\frac{1}{i^\ast \chi \otimes W_3(T Q)}}(Q) \stackrel{i_!}{\longrightarrow} C_{\frac{1}{\chi \otimes W_3(T X)}}(X) \,.$

If we redefine the twist on $X$ to absorb this “quantum correction” as $\chi \mapsto \frac{1}{\chi \otimes W_3(T X)}$ then this is

$C_{i^\ast \chi\frac{W_3(i^\ast T X)}{W_3(T Q)}}(Q) \stackrel{i_!}{\longrightarrow} C_{\chi}(X) \,,$

Postcomposition with this map in KK-theory now yields a map from the $i^\ast \chi \otimes W_3(N Q)$-twisted K-theory of $Q$ to the $\chi$-twisted K-theory of $X$:

$i_! \colon K_{\bullet + W_3(N Q) + i^\ast \chi}(Q) \to K_{\bullet +\chi} \,.$

This is the twisted Umkehr map in this context.

If we here think of $i \colon Q \hookrightarrow X$ as being the inclusion of a D-brane worldvolume, then $\chi$ would be the class of the background B-field and an element

$[\xi] \in K_{\bullet + W_3(N Q) + i^\ast \chi}(Q)$

is called (the K-class of) a Chan-Paton gauge field on the D-brane satisfying the Freed-Witten-Kapustin anomaly cancellation mechanism. (The orginal Freed-Witten anomaly cancellation assumes $\xi$ given by a twisted line bundle in which case it exhibits a twisted spin^c structure on $Q$.) Finally its push-forward

$[i_! \xi] \in K_{\bullet- \chi}(X)$

is called the corresponding D-brane charge.

See (Nuiten 13).

## Examples

###### Example

For $A = C_0(X)$ the algebra of functions on a compact complex manifold $X$, then $A$ is a PD algebra with fundamental class $\Delta$ in K-homology given by the Dolbeault operator on $X \times X$.

###### Example

For $A = C_0(X)$ the algebra of functions vanishing at infinity of a manifold $X$ with spin^c structure. Take $B = C_0(T^\ast X) \simeq_{KK} A^{op} \simeq A$. Then $\Delta$ constructed from the Dirac operator on the Clifford algebra bundle over $T^\ast X$ is a fundamental class.

## References

• Pascal Lambrechts, Don Stanley, Poincaré duality and commutative differential graded algebras (arXiv:math/0701309)

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

• Henri Moscovici, Eigenvalue inequalities and Poincaré duality in noncommutative geometry, Commun. Math. Phys. 184 , 3 (1997) 619

Chapter 6.4 $\beta$ (starting p. 601) in

Def. 2.1 in

Duality including the twisted K-theory induced by twisted spin^c structure over manifolds is discussed in section 7 of

and generalized to equivariant KK-theory in

More on dual objects in KK is in

Discussion of the twisted Umkehr map and the Freed-Witten-Kapustin anomaly in this context is in

Revised on August 19, 2013 16:30:08 by Urs Schreiber (24.131.18.91)