EXCERCISES, LIST 3 projections, balls, spheres, directed geodesics

1. Prove Link Lemma using Projection Lemma.

2. Show that if X is not finite dimensional then not necessarily (a) if dist(v_{0}, v) = 2 then Res(σ, X) ∩ B_{1}(v_{0}, X) is a simplex,

(b) if dist(v_{0}, v) = n then [B_{n}(v_{0}, X)]_{v} has a form B_{1}(τ, X_{v}) for some simplex τ. Check if these facts generalize by replacing simplices with cliques.

3. (Piotr Przytycki) Let Q_{1}, Q_{2} be convex subcomplexes in a 6-large simplicial complex X such that d_{X}(Q_{1}, Q_{2}) = 2. Prove that the intersection of 1-neighbourhoods B_{1}(Q_{1}) ∩ B_{2}(Q_{2}) is ∞-large.

4. (Piotr Przytycki) Show that there is a universal bound on the diameter of a regular triangulated flat hexagon (isometrically embedded in X) contained in a single sphere S_{n}(v, X).

5. Show that spheres in systolic pseudomanifolds are pseudomanifolds. Show that systolic pseudomanifolds (or the universal covers of 6-large pseudomanifolds) are unbounded. Show that the fundamental group of a finite 6-large pseudomanifold is infinite.

6. (Frederic Haglund) A “rhomb” modification of a geodesic is a replacement of some interior vertex with some adjacent vertex. Show that any two geodesics in X^{(1)} with common endpoints can be turned to one another by a sequence of rhomb modifications.

7. Prove one-sided fellow-traveller property for directed geodesics (using projection rays).

8. Show that fellow-traveller property does not hold for geodesics in X^{(1)} for a systolic complex X.

9. Show that geodesics in X^{(1)} with common endpoints, of length n, remain at Hausdorff distance ≤ \frac{n}{2}. Indicate examples when equality holds.

10. Show that directed geodesics from v to w, and from w to v, remain at Hausdorff distance ≤ \frac{n}{4}, where n = d_{X}(v, w).

11. (Frederic Haglund) Show that, given v_{0}, v_{1}, v_{2} with d(v_{0}, v_{1}) = d(v_{1}, v_{2}) = n and d(v_{1}, v_{2}) = d < n, in a systolic complex X, there are geodesics in X^{(1)} from v_{0} to v_{1} and from v_{0} to v_{2} which share initial part of length n − d.

12. Any full subcomplex of a systolic complex is aspherical. Hint: use contractibility of systolic complexes.

13. (Projection Lemma of Damian Osajda for 7-systolic complexes) Let X be a 7-systolic complex, and Q its convex subcomplex. Show that if v, w are adjacent vertices in S_{n}(Q, X), for some n ≥ 2, then the simplices being projections of v and w on B_{n−1}(Q, X) are incident, i.e. one is a face of the other. Show that this is in general not true in 6-systolic complexes.

14. Any directed geodesic is a projection ray.