A generalization of the morse complex
H. Blaine Lawson, Janko Latschev · 1998
Let $f:X\longrightarrow\IR$ be a Morse-Bott function on a compact manifold, whose gradient-like flow $\varphi\sb\tau$ satisfies a generalization of the Smale condition and is 'tame' near the critical manifolds. We show that such a flow satisfies the finite volume condition of Harvey and Lawson (HL97b). This implies that $\varphi\sb\tau$ gives rise to deformations of both the de Rham complex of differential forms on X and the complex of smooth singular chains transverse to the unstable manifolds of critical sets. We describe the structure of lim$\sb{\tau\longrightarrow}\varphi\sb\tau(T)$ for T in either of the two complexes. In particular, we show how the deformation of the singular chains yields an effectively computable model of the homology of X in terms of a generalized Morse complex (${\cal M},\partial\sb{f}$). The chain groups of this complex can be identified with the (suitably shifted) groups of singular chains in the critical sets, and the differential is explicitly given in terms of the flow. Applications include computations of the homology of a fibration and G-equivariant homology for manifolds with an action of a compact Lie group G. The methods also give partial results about the ring structure of $H\sp *(X)$.