98f:53034 53C21 53C20 Hicks, Andrew(1-PA)
Group actions and the topology of nonnegatively curved $4$-manifolds.
(English. English summary)
Illinois J. Math. 41 (1997), no. 3, 421--437.

Four-dimensional manifolds with nonnegative or positive sectional
curvature are still far from being understood. The Hopf conjecture
about the nonexistence of metrics with positive sectional curvature in
$S\sp 2\times S\sp 2$ remains unanswered. In 1989, W.-Y. Hsiang and
B. Kleiner obtained an important piece of information: any
four-dimensional manifold with positive curvature and with an $S\sp 1$
isometric action is homeomorphic to $S\sp 4$ or $\bold C{\rm P}\sp 2$
[J. Differential Geom. 29 (1989), no. 3, 615--621; MR 90e:53053].
In this paper, the author combines ideas from the above paper with a
new metric invariant due to Grove and Markovsen called the $q$-extent,
to prove the following two results: Theorem 1. If the isometry group
of a compact 4-dimensional manifold with nonnegative sectional
curvature contains $Z\sb p\times Z\sb p$ for $p$ large enough, then
the Euler characteristic of $M$ does not exceed 5. Theorem 2. If the
sectional curvature of $M\sp 4$ is pinched between $\delta$ and 1
(i.e. $1\ge \sec M\ge \delta$), and if the isometry group of $M$ is
large enough in terms of $\delta$, then the Euler characteristic of
$M$ does not exceed 3.

In fact, the author proves that the above sets of hypotheses limit
enormously the possible fixed point sets for the isometric actions,
thus obtaining the desired bounds on the Euler characteristic.

The paper is clearly written, and a nice combination of some of the
main tools used recently in the study of spaces with lower curvature
bounds.

Reviewed by Luis Guijarro

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