Output list
Preprint
Some counterexamples in surface homology
Published 04/25/2023
We present four counterexamples in surface homology. The first example shows that even if the loops inducing a homology basis intersect each other at most once, they still may separate the surface into two parts. The other three examples show some difficulties in working with minimal homology bases.
Preprint
Short homology bases for hyperelliptic hyperbolic surfaces
Published 06/14/2022
Given a hyperelliptic hyperbolic surface $S$ of genus $g \geq 2$, we find bounds on the lengths of homologically independent loops on $S$. As a consequence, we show that for any $\lambda \in (0,1)$ there exists a constant $N(\lambda)$ such that every such surface has at least $\lceil \lambda \cdot \frac{2}{3} g \rceil$ homologically independent loops of length at most $N(\lambda)$, extending the result in [Mu] and [BPS]. This allows us to extend the constant upper bound obtained in [Mu] on the minimal length of non-zero period lattice vectors of hyperelliptic Riemann surfaces to almost $\frac{2}{3} g$ linearly independent vectors.