Systolic freedom of 3-manifolds

Abstract

In this thesis, we study the Z2-coefficient homology (1, 2)-systolic freedom of 3-manifolds. In 1994, Bearard-Bergery and Katz proved the Z-coefficient homology (1, 2)-systolic freedom of S2 x S1. More generally, compact and orientable 3-manifolds are of Z-coefficient homology (1, 2)-systolic freedom due to the work of Babenko and Katz. Later in 1999, Freedman showed that S2 x S1 is of Z2-coefficient homology (1, 2)-systolic freedom, which is a counterexample to Gromov's conjecture. In the thesis, we show that the 3-manifold RP3 # RP3 is of Z2-coefficient homology (1, 2)-systolic freedom. The proof is based on the semibundle structure property of RP3 # RP3 and the application of Freedman's technique on S2 x S1. We show the details of how Dehn surgery changes metric on mapping torus in Freedman's example. Then with respect to the sequence of metrics constructed, we calculate the lower bound estimates of Z2-coefficient homology 1-systole and Z2-coefficient homology 2-systole, as well as the upper bound estimates of the volume of S2 x S1 in details. The 3-manifold RP3 # RP3 has a sphere semibundle structure. We employ Freedman's technique to construct a sequence of Riemannian metrics on RP3 # RP3. By an investigation of 3-manifolds with semibundle structure, we prove a lower bound estimate of Z2-coefficient homology 1-systole of RP3 # RP3. A lower bound estimate for Z2-coefficient homology 2-systole of RP3 # RP3 is obtained in terms of the semibundle structure and Freedman's result on S2 x S1. Based on these estimations, we prove the Z2-coefficient homology (1, 2)-systolic freedom of RP3 # RP3.