The first integral cohomology of pure mapping class groups

Abstract

It is a classical result of Powell that pure mapping class groups of connected, orientable surfaces of finite type and genus at least three are perfect. In stark contrast, we construct nontrivial homomorphisms from infinite-genus mapping class groups to the integers. Moreover, we compute the first integral cohomology group associated to the pure mapping class group of any connected orientable surface of genus at least 2 in terms of the surface's simplicial homology. In order to do this, we show that pure mapping class groups of infinite-genus surfaces split as a semi-direct product.

Figures (5)

• Figure 1: The circles are identified vertically to obtain $\Sigma$.
• Figure 2: On the two-ended infinite-genus surface with no planar ends $L$, there is a unique nontrivial simple separating homology class $v$. Shown here is a---portion of a---$\mathbb Z$-coloring of $\mathcal{C}_v(L)$. Two of the curves shown are (necessarily) colored by 0.
• Figure 3: $R, \Sigma, \gamma, \gamma_1, \gamma_2, \partial_0$ as in the proof of Lemma \ref{['lem:additive']}.
• Figure 4: An example of $K_1$ (red) sitting in $K_2$ and the corresponding basis elements.
• Figure 5: The surface $Z_n$.

Theorems & Definitions (35)

• Theorem 1
• Corollary 2
• proof
• Corollary 3
• proof
• Theorem 4
• Theorem 5
• Corollary 6
• Corollary 7
• Definition 2.1