Lifts of simple curves in finite regular coverings of closed surfaces

Abstract

Suppose S is a closed orientable surface andS is a finite sheeted regular cover of S. When studying mapping class groups, the following question arose: Do the lifts of simple curves from S generate H 1 (S, Z)? A family of examples is given for which the answer is "no".

Quotients of surface groups and homology of finite covers via quantum representations

“…Again, Theorem 1.2 holds for all nonabelian surface groups, and in the abelian case the result obviously cannot hold. We remark that I. Irmer has proposed a version of Theorem 1.2 in [15], and she proves that H s 1 (S , Z) = H 1 (S , Z) whenever the deck group S → S is abelian.…”

Generating the homology of covers of surfaces

“…Part (ii) is a theorem of Irmer [6, Lemma 6]. We will give a simplified version of her argument below that avoids most of its complicated combinatorial group theory.…”

“…However, Theorem C suggests that the answer to this should be “no”. Remark An important ingredient in our proof of Theorem C is a theorem of Irmer [6] giving certain finite abelian covers $$\pi \colon \widetilde{\Sigma }\rightarrow \Sigma$$ for which $$\operatorname{H}_1^{\sigma }(\widetilde{\Sigma };\mathbb {Z})$$ is a proper subgroup of $$\operatorname{H}_1(\widetilde{\Sigma };\mathbb {Z})$$ (see Theorem 5.2(ii)). To make this paper more self‐contained, we also include a simplified proof of this theorem.…”

Finding simple curves in surface covers is undecidable

“…Another source of covers without simple curves comes from Section 3.5 of [5]. In [5], examples of covering spaces are given, for which connected components of pre-images of simple curves do not span the integer homology of the covering space. These covering spaces are constructed by iterating mod m homology covers, as will now be explained briefly.…”

“…However, the only simple curves in mod m homology covers are contained in [π 1 (S), π 1 (S)], because by Lemma 3, none of the other elements of π 1 (S) are contained in primitive homology classes in H 1 (S; Z). As shown in Lemma 3 of [5], none of the simple curves in [π 1 (S), π 1 (S)] are contained in [π 1 (S), π 1 (S)], so any simple curves in a mod m homology coverS are nonseparating inS. Taking a mod m homology coverS ofS therefore kills off any simple curves that were inS.…”