n
Topology of Hyperbolic Actions of R on n-manifolds
Damien Bouloc
Institut de Mathématiques de Toulouse, Université Paul Sabatier.
[email protected] — http://www.math.univ-toulouse.fr/~dbouloc/
Classification of Totally Hyperbolic Actions
n
A totally hyperbolic action of R on a n-manifold M
n
is a nondegenerate action ρ : R × M → M which
is faithful. Its n-dimensional orbits are called hyperbolic domains, they are homeomorphic to polytopes
in Rn .
A hyperbolic action ρ is determined (up to isomorphism) by closed hypersurfaces H1 , . . . , HN in M and
vectors v1 , . . . , vN in Rn such that:
1. the hyperbolic domains are the connected components of M \ ∪i Hi ,
2. for any p ∈ M , limt→∞ ρ(−tvi , p) lie in Hi ,
3. the hypersurfaces Hi1 , . . . , Hik bounding a
given domain have the same layout as the vectors vi1 , . . . , vik in Rn , and the latter form a
complete fan of Rn .
Example in Dimension 3
Conversely, if a family of hypersurfaces H1 , . . . , HN
splits M into curved polytopes, and can be associated
to a family of vectors v1 , . . . , vN satisfying Condition
3, then it can be realized by an hyperbolic action on
M.
We use the decomposition of the sphere S 3 = {|z1 |2 + |z2 |2 = 1} into two solid tori T1 and T2 .
H1
v2
H2
w
v1
v3
H5
H3
H4
−w
v4
v5
H0
H1
H2
H1
H3
H2
H4
H1
H3
H2
H4
H3
H1
v1
H5
H2
v5
v2
H4
v4
T1
v3
T2
T1
This example is invariant by the involution σ : (z1 , z2 ) 7→ (−z1 , −z2 ), so it also determines an hyperbolic
action on the projective space RP 3 = S 3 /σ.
Morse Theory
For a fixed w ∈ Rn , define the flow ϕtw = ρ(−tw, .). If
p0 is a fixed point at the intersection of Hi1 , . . . , Hin ,
we have the following normal form: there are local
coordinates (x1 , . . . , xn ) centered at p0 such that
ϕtw (x1 , . . . , xn ) = (x1 e−α1 t , . . . , xn e−αn t )
where (α1 , . . . , αn ) are the coordinates of w in the
basis (vi1 , . . . , vin ). We set
Indp0 (w) = card{1 ≤ i ≤ n | αi > 0}
the number of attractive directions of ϕtw around p0 .
H4
H4
T2
S2
H0
H3
Number of Hyperbolic Domains
(res. repulsive) fixed point for ϕtw . This implies that
the number of hyperbolic domains of ρ is equal to
n
n
D = c0 2 = cn 2
where ci denotes the number of fixed points p of index Indp (w) equal to i. In particular, c0 = cn does
not depend on w.
Theorem
For any generic w, the fixed points of ρ are the
singular point of a Morse function f : M → R
such that the Morse index of a singular point p is
exactly Indp (w).
Adding to the family of hypersurfaces Hi
two concentric spheres around a fixed point
create a new decomposition of M that is
still valid. It follows that if M admits a
hyperbolic action with D domains, it also
admits a hyperbolic action with D + 2n+1
domains.
In particular, in
– the sphere
– any closed
– any closed
H1
H1
S
S0
H2
v3
H2
v1
v0
v
v...
v2
dimension 2, for any k ≥ 1,
S 2 admits a hyperbolic action with 8k domains,
orientable surface Σg of genus g > 0 admits a hyperbolic action with 4k domains,
non-orientable surface Σ0g of genus g ≥ 0 admits a hyperbolic action with 4k domains.
We can then apply Morse inequalities to obtain:
ci − ci−1 + · · · + (−1)i c0 ≥ bi − bi−1 + · · · + (−1)i b0
If w is generic, that is not colinear to any vi , then on where bi are the Betti numbers of M . When i = n,
each hyperbolic domain there is a unique attractive it is an equality and the RHS term is χ(M ).
References
— N.T. Zung, N. Van Minh, Geometry of nondegenerate Rn -actions on
n-manifolds, J. Math. Soc. Japan (2014)
— D. Bouloc, Some remarks on the topology of hyperbolic actions of Rn on
n-manifolds, arXiv (to appear)
Action with 4 domains on Σg
(g > 0)
Action with 8 domains on Σg
(quotients to 4 domains on Σ0g )
Action with 16 domains on Σg
(quotients to 8 domains on Σ0g )
Recall that D = 4c0 = 4cn , so equivalently, for any k ≥ 1,
– the sphere S 2 admits a hyperbolic action such that c0 = cn = 2k,
– any other closed surface admits a hyperbolic action such that c0 = cn = k.
Why is cn even on the sphere ? The index Indp (w) of p ∈ Hi ∩ Hj depends only on the position of w
with respect to vi and vj . In particular, all the intersection points between two closed loops Hi and Hj are
together attractive or not attractive. But on the sphere, two loops intersect an even number of times.