"Universal secondary characteristic homomorphism of pairs of regular Lie Algebroids"

    There are three theories of secondary characteristic classes on Lie algebroids, first was given by J.Kubarski [1991], the next was given by R.L.Fernandes [2002] and the last by M.Crainic [2003]. These theories generalize the classical secondary characteristic classes for principal bundles and foliations.

    The purpose of this lecture is to present a universal secondary characteristic homomorphism

 

            h_(B,A) ::H(g,B)→H(A)

 

 for a pair (B,A), BÌA, of regular Lie algebroids (nonflat in general) over the same foliated manifold (M,F) [especially for transitive ones] where g= ker#_A .

    This homomorphism h_(B,A) has the following property:

    --- for an arbitrary (nonregular in general) Lie algebroid L on M and for a flat L-connection in A (i.e. a homomorphism of Lie algebroids) Ñ:L→A, the superposition

 

            Ñ^#o h_(B,A): H(g,B)→H(L)

 

 describes --- for the suitable (B,A, Ñ) ---

    a) the classical secondary "flat" characteristic homomorphism for a principal bundle with a reduction and flat connection, see for example papers by Kamber-Tondeur;

    b) the Crainic characteristic classes for a representation of L on a vector bundle.