PERIODIC HAMILTONIAN FLOWS ON FOUR MANIFOLDS 3
The results of this paper for the case of isolated fixed points already
appeared in the author's thesis [Kl]. A five page summary of this paper
appeared in [K2].
S. Tolman and the author are currently writing up a treatment of
Hamiltonian torus actions on higher dimensional manifolds where the
dimension of the torus is one less than half the dimension of the man-
ifold [KT].
Acknowledgements. I wish to thank the following people for
useful discussions: D. Abramovich, M. Audin, A. Canas Da Silva,
V. L. Ginzburg, M. Grossberg, V. Guillemin, J. Harris, D. McDuff,
E. Lerman, R. Sjamaar, S. Sternberg, S. Tolman, C. Woodward. The
foundations of the subject lie in early papers by Guillemin and Stern-
berg and by Atiyah [A, GS1]. I was influenced by the papers of M.
Audin [Aul, Au2], K. Ahara and A. Hattori [AH], and T. Delzant
[De]. M. Audin, E. Lerman, S. Singer, and S. Tolman, made helpful
remarks on various versions of the manuscript. The referee made excel-
lent suggestions for improving the exposition, which lead to a complete
revision of the paper, for the better.
I proceed with more specific acknowledgements:
Many ideas come from [Aul, Au2, AH]: the chains of gradient
spheres appear there, Lemma 5.16 is very similar to [Aul, §3.3] (the
proof is different), Proposition 5.13 was claimed in [Aul, Lemma 3.1.2]
and was proved in [AH, §6] (our proof is different), subsection 6.2 con-
tain a new proof of results of [Aul, AH]. Section 4 is strongly influenced
by the first section of Delzant's paper, [De]. D. McDuff helped me un-
derstand the work of Audin, Ahara and Hattori, she pointed out the
problem with [Aul, Prop.3.1.2], and she indicated that one only needs
to blow down and not to blow up, which made life and section 6 sim-
pler. S. Tolman convinced me to study Hamiltonian circle actions in
dimension four before attempting to understand them in higher dimen-
sion; if not for her advice I would still be staring in space. She clarified
the important Example 3.10, and pointed at gaps in an earlier proof of
Proposition 4.3. I learned of Nakai's criterion from D. McDuff and L.
Polterovich, and I could not have applied it in section 7 without con-
sulting D. Abramovich and J. Harris. I learned the trick in the proof
of Lemma 5.10 from a talk of A. Khovanskii, and I learned Lemma 5.4
from M. Grossberg. Proposition 4.11 was conjectured by T. Delzant
and V. Ginzburg, from whom I heard this conjecture.
I wish to thank the Weizmann Institute of science for their hospitality
and support during the summer of 1992. I was supported by an Alfred
P. Sloan Dissertation Fellowship in the academic year 1992-93. I was
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