A normal generating set for the Torelli group of a non-orientable closed surface

Abstract

For a closed surface S, its Torelli group I(S) is the subgroup of the mapping class group of S consisting of elements acting trivially on H 1 (S; Z). When S is orientable, a generating set for I(S) is known (see [13]). In this paper, we give a normal generating set of I(N g ) for g ≥ 4, where N g is a genus-g non-orientable closed surface.

A normal generating set for the Torelli group of a compact non-orientable surface

“…For example, in Theorem 1.1, t β t −1 β ′ is a BP map. Hirose and the author [5] obtained the following theorem.…”

“…We prove ). At first, I(N g ) is normally generated by t α , t β t −1 β ′ and t γ (see [5]). Hence we have that I(N 1 g ) is normally generated by t α , t β t −1 β ′ , t γ and t δ 1 , t ρ 1 .…”

“…Hence we obtain a normal generating set for I(N b g ). In particular, for g ≥ 5 since I(N g ) is normally generated by t α and t β t −1 β ′ (see [5]), we do not need t γ as a normal generator of I(N b g ) for g ≥ 5. Finally, proving the following lemma, we finish the proof of Theorem 1.1.…”

“…In particular, Johnson [6] showed that the Torelli group of an orientable closed surface is finitely generated by BP maps. Hirose and the author [5] showed that I(N g ) is generated by BSCC maps and BP maps for g ≥ 4. In this paper, we give an explicit normal generating set for I(N b g ) consisting of BSCC maps and BP maps, for g ≥ 4 and b ≥ 1.…”

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For a compact surface S, let I(S) denote the Torelli group of S. For a compact orientable surface Σ, I(Σ) is generated by BSCC maps and BP maps (see [10] and [11]). For a non-orientable closed surface N , I(N ) is generated by BSCC maps and BP maps (see [5]). In this paper, we give an explicit normal generating set for I(N b g ), where N b g is a genus-g compact non-orientable surface with b boundary components for g ≥ 4 and b ≥ 1.

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