Given a manifold, the vertices of a geometric intersection graph are defined as a class of submanifolds. Whether there is an edge between two vertices depends on their geometric intersection numbers. The geometric intersection complex is the clique complex induced by the geometric intersection graph. Common examples include the curve (arc) complex and the Kakimizu complex. Curve complexes and arc complexes are used to understand mapping class groups and Teichmüller spaces, while Kakimizu complexes are primarily used to study hyperbolic knots. We can study these geometric intersection complexes from various perspectives, including topology (e.g., homotopy type), geometry (e.g., dimension, diameter, hyperbolicity), and number-theoretic connections (e.g., trace formulas of maximal systems). In this talk, we will mainly explain how to determine the dimension of the (complete) $1$-curve (or arc) complex on a non-orientable surface and examine the transitivity of maximal complete $1$-systems of loops on a punctured projective plane.
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