Orientation

Abstract

Summing up the analytical foundational results we have developed up to this stage, we notice that this knowledge about the trajectory spaces of the time-independent and time-dependent negative gradient flow enables us already to build a Morse homology theory with coefficients in the field ℤ₂. However, in order to admit arbitrary coefficient groups, i.e. coefficients in ℤ, we still have to accomplish more elaborate results concerning the characteristic intersection numbers for the unparametrized trajectories. Referring to the introduction, we may deduce these intersection numbers from a comparison of the canonical orientation of the intersection manifold $W^u (x) \cap |W^s \left( y \right) \approx \mathcal{M}_{x,y}$ by the negative gradient field with some coherent orientation related to the critical points.