There has been recent interest in determining which sets of curves on a possibly punctured surface S, when lifted, generate the homology of every regular finite cover of S; there is an analogous question for regular finite covers of graphs. Work of Malestein and Putman shows that for any punctured S of sufficient genus, the rational homology of some regular finite covers of S is not generated by the lifts of curves in any finite union of Aut(F_n)-orbits. This implies that there are regular finite covers of finite graphs H whose rational homology is not generated by lifts of primitive elements in the fundamental group of H, answering a question of Farb and Hensel; it also implies that the rational homology of finite covers of punctured surfaces is not always generated by lifts of curves with at most k self-intersections (e.g., simple closed curves). Recent work of Boggi, Putman, and Salter shows that for closed surfaces, lifts of "pants curves" generate the rational homology of every finite branched cover. We are focused on the following question: do lifts of curves conjugate into a proper free factor of the free group of rank n generate the fundamental group of every regular finite cover of the n-rose? This question is related to the congruence subgroup problem for the fundamental group of a genus-two surface due to work of Boggi and unpublished work of Kent. During the talk, we will show that there are naturally-defined spaces Wh(G) associated to each regular finite cover G of the rose whose connectedness would imply an affirmative answer to this question for G. All such Wh(G) are k-dense (for some k) in the "Whitehead space of the rose," a connected space with an action of Out(F_n) by isometries. We will define these spaces, discuss why the connectedness of Wh(G) for all G would imply an affirmative answer to our question above, and describe some of the topological properties of Wh(G) and the Whitehead space of the rose.
Homology of regular finite covers of graphs
Becky Eastham, University of Wisconsin-Madison
2023 AWM Research Symposium
Women in Groups, Geometry, and Dynamics [Organized by Hannah Hoganson and Rylee Lyman]