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For A a dg (or A-infinity) algebra and M a module over A, we study the imageof the characteristic morphism $\chi_M: HH^*(A, A) \to Ext_A(M, M)$ and itsinteraction with the higher structure on the Yoneda algebra $Ext_A(M, M)$. Tothis end, we introduce and study a notion of A-infinity centre for minimalA-infinity algebras, agreeing with the usual centre in the case that there isno higher structure. We show that the image of $\chi_M$ lands in the A-infinitycentre of $Ext_A(M, M)$. When A is augmented over k, we show (under mildconnectedness assumptions) that the morphism $\chi_k: HH^*(A, A) \toExt_A(k,k)$ into the Koszul dual algebra lands exactly onto the A-infinitycentre, generalising the situation from the Koszul case established byBuchweitz, Green, Snashall and Solberg. We give techniques for computingA-infinity centres, hence for computing the image of the characteristicmorphism, and provide worked-out examples. We further study applications totopology. In particular we relate the A-infinity centre of the Pontryaginalgebra to a wrong way map coming from the homology of the free loop space,first studied by Chas and Sullivan.

The A-infinity Centre of the Yoneda Algebra and the Characteristic Action of Hochschild Cohomology on the Derived Category