MathDB

Let

M

be a closed, orientable

3

-dimensional differentiable manifold, and let

G

be a finite group of orientation preserving diffeomorphisms of

M

. Let

P

and

Q

denote the set of those points of

M

whose stabilizer is nontrivial (that is, contains a nonidentity element of

G

) and noncyclic, respectively. Let

\chi (P)

denote the Euler characteristic of

P

. Prove that the order of

G

divides

\chi (P)

, and

Q

is the union of

-2\frac{\chi(P)}{|G|}

orbits of

G

Problem Statement