related discussion on MO here

]]>I have added at *K-orientation and Umkehr maps* discussion of push-forward in twisted K-theory.

Essentially the same discussion I have (with slight editorial changes) also copied to

Also added more related references to all these entries.

]]>added to *Poincare duality algebra*

two more references on the $C^\ast$-case;

remarks highlighting that the $C^\ast$-case is about duality in K-theory as opposed to in ordinary cohomology;

a remark that for twisted groupoid convolution algebras the fact that the Poincaré duality is to the

*opposite*algebra corresponds to keeping the same groupoid (up to equivalence) but inverting the twist(this is no hard to see, but if anyone knows a place where this is remarked in the literature, I’d be grateful for a pointer.)

okay, I have added the definition *For graded-commutative algebras*.

Let me know your favorite reference, and I’ll add it, too.

]]>surely there are many other Poincaré duality algebras

jim ]]>

I am starting an entry *Poincaré duality algebra*, but it still needs some attention