Funtoriality and duality in Morse-Conley-Floer homology

In [J. Topol. Anal. 6 (2014), 305–338], we have developed a homology theory (Morse–Conley–Floer homology) for isolated invariant sets of arbitrary flows on finite-dimensional manifolds. In this paper, we investigate functoriality and duality of this homology theory. As a preliminary, we investigate functoriality in Morse homology. Functoriality for Morse homology of closed manifolds is known, but the proofs use isomorphisms to other homology theories. We give direct proofs by analyzing appropriate moduli spaces. The notions of isolated map and flow map allow the results to generalize to local Morse homology and Morse–Conley–Floer homology. We prove Poincaré-type duality statements for local Morse homology and Morse–Conley–Floer homology.