Well definedness of square roots of separating Dehn Twists

Question

Let $c_1,c_2,c_3,c_4,c_5$ be a five chain of circles on a genus 2 surface (i.e $i(c_k,c_{k+1})=1$ and zero otherwise). Then $(T_{c_1} T_{c_2})^6 = (T_{c_4} T_{c_5})^6 = T_c$ where $c$ is a separating curve that bounds a neighborhood of $c_1$ and $c_2$ (and also bounds a neighborhood of $c_4$ and $c_5$). This follows from the two chain relation. Then we can think of $(T_{c_1} T_{c_2})^3$ and $(T_{c_4} T_{c_5})^3$ as square roots of $T_c$. Are these two equal as mapping classes?

Answer 1

They are different. In fact, they act differently on $H_1(\Sigma_2;\mathbb{Z})$. Let $V \subset H_1(\Sigma_2;\mathbb{Z})$ be the span of the homology classes of $c_1$ and $c_2$, and let $W \subset H_1(\Sigma_2;\mathbb{Z})$ be the span of the homology classes of $c_4$ and $c_5$. You can then calculate the $(T_{c_1} T_{c_2})^3$ acts by $-1$ on $V$ while fixing $W$, but $(T_{c_4} T_{c_5})^3$ acts by $-1$ on $W$ while fixing $V$.

There is by now a large literature on what kinds of roots a Dehn twist has, starting with

D. Margalit, S. Schleimer Dehn twists have roots. Geom. Topol. 13 (2009), no. 3, 1495-1497.

and continuing with

D. McCullough, Darryl, K. Rajeevsarathy, Kashyap Roots of Dehn twists. Geom. Dedicata 151 (2011), 397-409.

and

K. Rajeevsarathy Roots of Dehn twists about separating curves. J. Aust. Math. Soc. 95 (2013), no. 2, 266–288.

and

N. Monden On roots of Dehn twists. Rocky Mountain J. Math. 44 (2014), no. 3, 987–1001.