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Axioms 2015, 4(3), 313-320; doi:10.3390/axioms4030313
Abstract: I discuss old and new results on fixed points of local actions by Lie groups G on real and complex 2-manifolds, and zero sets of Lie algebras of vector fields. Results of E. Lima, J. Plante and C. Bonatti are reviewed.
Every flow on a compact manifold of non-zero Euler characteristic has a fixed point.
Here the Lie group is the group of real numbers.
The earliest papers I have found on fixed points for actions of other non-discrete Lie group are those of P. A. Smith  (1942) and H. Wang  (1952). Then came Armand Borel’s landmark paper of 1956:
(Borel ). If H is a solvable, irreducible affine algebraic group over an algebraically closed field , every algebraic action of H on a complete algebraic variety over has a fixed point.
Over the field of complex numbers, completeness is equivalent to compactness in the classical topology, and complete nonsingular varieties are compact Kähler manifolds.
In 1973, A. Sommese  extended Borel’s theorem to solvable holomorphic actions on compact Kähler manifolds with first Betti number 0. In contrast to the results below, these have no explicit restrictions on dimensions or Euler characteristics.
2. Actions and Local Actions
If denotes a map, its domain is and its range is .
Let denote maps. Regardless of their domains and ranges, the composition is defined as the map whose domain, perhaps empty, is . The associative law holds for these compositions: The maps and have the same domain
Henceforth M denotes a manifold with boundary ∂M, and G denotes a connected Lie group with Lie algebra 𝔤.
A local homeomorphism f on M is a homeomorphism between open subsets of M. The set of these homeomorphisms is denoted by .
A local action of G on M is a triple , where is a function having the following properties:
The set is an open neighborhood of .
The evaluation map
is the identity map of M.
The maps and agree on the intersection of their domains.
When the local action is a global action. If G is simply connected and M is compact, every local action extends to a unique global action.
When α has been specified, we define the fixed-point sets
A local flow is a local action . In this case we set and identify Ψ with the indexed family of of local maps in M. If is a local action, to every there corresponds a local flow defined in the following. Consider X as a 1-parameter subgroup of G, i.e., a homomorphism , and set . The local flow induced by a vector field X on M tangent to ∂M is denoted by .
A block for a local flow Ψ (a Ψ-block) is a compact having a precompact open neighborhood , termed isolating, such that . When this holds, the index of Ψ in U is defined as the fixed point index of for sufficiently small , as defined by Dold  (see also Brown  and Granas and Dugundji ). This integer depends only on K, and we set . When then K is essential. If K is a block for the local flow of a vector field X, an equivalent definition of as the Poincaré–Hopf index of X at K is given in Section 4.
3. Fixed Points of Local Actions on Surfaces
In the rest of this section M denotes a real closed surface (compact with empty boundary) and G is a connected Lie group acting continuously on M.
An important role is played by the group , the solvable group of real, upper triangulable matrices with positive diagonal entries. In his pioneering 1964 paper, E. Lima  constructed fixed-point free actions of on the compact 2-cell and the 2-sphere, but he also showed that every abelian Lie group action on a compact surface M of nonzero Euler characteristic has a fixed point. These results were extended in 1986 by Plante:
(Plante ). Let M be a compact surface whose boundary may be nonempty.
has a fixed-point free action on M.
If , every action on M by a connected nilpotent Lie group has a fixed point.
Many facts about existence of fixed points for continuous actions on closed surfaces can be derived from the results of M. Belliart summarized in the following theorem. If denotes a group of matrices, denotes the quotient of H by its center.
(Belliart ). There is a fixed-point free action of G on M iff one of the following conditions (a), (b), (c) holds:
and G is solvable but not nilpotent.
and G has as a quotient.
, G is semisimple, and either:
G has as a quotient, or
, , and G has as a quotient one of the groups
A Lie algebra is supersolvable if it is faithfully represented as upper triangular real matrices. A Lie group is supersolvable if its Lie algebra is.
has an effective analytic action on M.
If G has an effective, fixed-point free analytic action on M, then , with equality when G is a supersolvable and .
The following result gives upper and lower bounds on the number of fixed points of analytic actions of :
(Hirsch , Cor. 17, Thm 22).
Let M have genus g. For every there is an effective analytic action β of on M such that:
If G is not supersolvable and has an effective analytic action on M,
Can the right hand side of Equation (1) can be lowered?
4. Indices of Vector Fields
Let denote the vector space of vector fields (continuous sections of the tangent bundle) on a smooth manifold M, endowed with the compact open topology.
The zero set of is
A block for X (an X-block) is a compact, relatively open set . Every sufficiently small open neighborhood of K is isolating for X, meaning its closure is compact and . This implies that U is isolating for every vector field Y sufficiently close to X.
Let K be an X-block. When K is finite, the Poincaré–Hopf index of X at K, and in U, is the integer defined as follows. For each choose an open set meeting K only at p, such that W is the domain of a chart
Noting that , we define as the degree of the map defined for any sufficiently small as
The index of an arbitrary X-block K is the integer defined as the Poincaré–Hopf index of any sufficiently close approximation to X having only finitely many zeros in U .
This number is independent of U and is stable under perturbations of X. The X-block K is essential when . This implies because every isolating neighborhood of K meets .
(Poincaré–Hopf). If M is compact, for all continuous vector fields X on M.
This implies certain local actions of 2-dimensional abelian Lie groups have fixed points. The results below are analogs for local actions of nonabelian Lie groups.
(Hirsch ). Let M be a real surface, perhaps non-compact or having non-empty boundary. Let G be a connected nilpotent Lie group and an effective local action. Assume given a continuous local action of G on M, and let K be an essential block for the local flow induced by a 1-parameter subgroup. Then .
This implies Plante’s result, Theorem 3(ii).
Let and X be as in Theorem 8.
If is a compact attractor for and , then .
If has n essential blocks, then has n components.
The counter-example in Theorem 3(i) show that fixed point results for broader classes of Lie groups, including supersolvable groups, need stronger hypotheses.
Henceforth M denotes either a real or complex 2-manifold, the corresponding ground field being or . Let denote the Lie algebra of vector fields on M that are analytic over . If , denotes the induced local flow on the tangent vector bundle of M.
Assume . We say that Y tracks X if there exists a continuous map
Let denote a Lie algebra of vector fields. We say that tracks X provided each tracks X.
If X spans an ideal in then tracks X, and the converse holds if is finite dimensional.
The set is a Lie algebra that tracks X.
Assume , K is an essential X-bloc, and tracks X. Let one of the following conditions hold:
M is complex,
M is real and is supersolvable.
Here is a simple example in which the hypotheses hold. For M take complex projective 3-space. Let G be the solvable complex Lie group of unimodular 4× 4 upper triangular complex matrices. The natural action of G on induces an effective holomorphic action of G on M, mapping the Lie algebra of G isomorphically onto a Lie algebra . Let have the block in its upper right hand corner and all other elements equal to zero. X spans an ideal, the triple commutator subalgebra . The X-block , a copy of , is essential because ; and is a singleton in .
Conflicts of Interest
The author declares no conflict of interest.
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