By an m-function we mean a function on a manifold with boundary
having only non-degenerate critical points, no critical points near
the boundary and such that its restriction to the boundary has only
non-degenerate critical points. This is one of possible
generalizations of Morse functions to the case of manifolds with
boundary, the other one assumes that on each component of the
boundary the function is constant. The study of such functions was
started by Jankowski and Rubinsztein [4], Braess
[2]. Later the Morse - Smale theory for such type of
functions was developed in [3]. The interest for this
theory was revived by applications to monopole theory
[5]. In [1] the results of
[4,2,3] were rediscovered and detailed
analytical proofs were provided.
I will discuss the aspects of the theory of m-functions which seem
to be most interesting nowadays. In particular, I will show that the
results of [3] give the Morse inequalities for m-functions
improving those obtained in [6] by taking into account
some mysterious homology operations. I will discuss also another
basic method introduced in [3] of replacing a critical
point in the interior by two critical points on the boundary. These
ideas give the base for the calculations of minimal numbers of
critical points of m-functions for simply connected manifolds of
dimension at least 6, which are the main results of [3].
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References- M. Borodzik, A. Némethi, A. Ranicki, Morse
theory for manifolds with boundary , arXiv:1207.3066v3 [math.GT]
- D. Braess, Morse - Theorie fur berandete
Mannigfaltigkeiten , Math. Ann. 208, 1974, 133--148.
- B. Hajduk, Minimal m-functions ,
Fund. Math. 111, 1981, 179--200.
- A.Jankowski, R.Rubinsztein, Functions with
non-degenerate critical points on manifolds with boundary ,
Comm. Math. 16, 1972, 99--112.
- P. Kronheimer, T. Mrowka, Monopoles and three
manifolds , New Mathematical Monographs, 10. Cambridge
University Press, Cambridge, 2007.
- F. Laudenbach, A Morse complex on manifolds
with boundary , Geom. Dedicata 153, 2011, 47--57.
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