Transport functions for principal bundles and Morse homology with differential graded coefficients

Abstract

We study transport functions as a Morse-theoretical way of describing principal bundles. Transport functions are maps from the spaces of broken gradient flow lines to a topological group and they encode the transition functions of the principal bundle. We describe and extend a construction by Voigt that yields such transport functions and show that one can recover the principal bundle from the transport function. Using transport functions with values in a topological group $G$ and a differential graded module over the chains of $G$ we define a chain complex in the style of Barraud-Damian-Humili\`ere-Oancea's Morse homology with differential graded coefficients. We prove that in many cases the homology of this complex is the homology of an associated bundle. In the case of smooth bundles transport functions arise also from parallel transport with respect to a connection and the corresponding DG Morse complex turns out to be isomorphic to a complex defined in the style of Barraud-Damian-Humili\`ere-Oancea. We eventually consider certain aspects of the functoriality of our constructions.