Abstract
In this chapter, to a compact manifold endowed with a Morse function a pseudo-gradient field satisfying the Smale condition, we associate a complex. It is generated by the critical points of the function, and the differential is defined by the trajectories connecting critical points. We prove that this is indeed a complex, and that its homology does not depend on the actual function and vector field we used.
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Notes
- 1.
For this algebraic formalism, see Section B.1 and the references given there.
- 2.
Here we use vector spaces over Z/2; the correct general notion uses modules over a commutative ring, which includes the case of abelian groups. See Section 3.3.
- 3.
This figure is the very core of the proof.
- 4.
We adopt this term to translate the elegant “variété à bord anguleux” introduced by François Latour [46] for “with boundary and corners”.
- 5.
See Exercise 14 on p. 78, if necessary.
References
Floer, A.: Witten’s complex and infinite dimensional Morse theory. J. Differ. Geom. 30, 207–221 (1989)
Latour, F.: Existence de 1-formes fermées non singulières dans une classe de cohomologie de Rham. Publ. Math. IHÉS 80, 135–194 (1994)
Laudenbach, F.: Symplectic geometry and Floer homology, pp. 1–50. Sociedade Brasileira de Matemática (2004)
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Exercises
Exercises
Exercise 13
What is the homology of the complex associated with the function defined in Exercise 12 (p. 51) and the vector field suggested in the same exercise?
Exercise 14
Let E and F be two vector subspaces of a finite-dimensional real vector space. Show that an orientation of E is an equivalence class of bases of E for the equivalence relation
Likewise, verify that the relation
defines an equivalence relation on the bases of the complements of F that does not depend on the chosen basis 
of F. The equivalence classes are the co-orientations of F.
Verify that if E is oriented, F is co-oriented and E and F are transversal, then E∩F is co-oriented.
Exercise 15
Determine the homology of the complex (C ⋆(f;Z),∂ X ) for the examples of Morse functions on the manifolds P n(C), T 2 and S n used in this book.
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Audin, M., Damian, M. (2014). The Morse Complex. In: Morse Theory and Floer Homology. Universitext. Springer, London. https://doi.org/10.1007/978-1-4471-5496-9_3
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DOI: https://doi.org/10.1007/978-1-4471-5496-9_3
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