Connected-Sum Decompositions of Surfaces with Minimally-Intersecting Filling Pairs

TL;DR

This paper generalizes a construction for decomposing surfaces with minimally-intersecting filling pairs into connected sums, providing explicit algebraic methods to determine the resulting surface's homeomorphism class.

Contribution

It introduces a generalized, algebraic construction for connected-sum decompositions of surfaces with minimally-intersecting filling pairs, along with criteria for such decompositions.

Findings

Explicit algebraic method for determining homeomorphism class

Generalization of previous construction for surface decomposition

Criteria for decomposing surfaces with minimally-intersecting filling pairs

Abstract

Let $ S_g $ be a closed surface of genus $ g $ and let $ (\alpha, \beta) $ be a filling pair on $ S_g $; then $ i(\alpha, \beta) \geq 2g-1 $, where $ i $ is the (geometric) intersection number. Aougab and Huang demonstrated that (exponentially many) minimally-intersecting filling pairs exist on $ S_g $ when $ g > 2 $ by a construction which produces higher-genus surfaces with filling pairs as connected sums of lower-genus surfaces with filling pairs. We present a generalization of their construction which provides an explicit, algebraic means of determining the homeomorphism class of the resulting pair, and a criterion for determining when a surface with minimally-intersecting filling pair admits a decomposition as a connected sum.