## Notes of Vlad Markovic’s second Cambridge lecture 22-06-2017

Homology of curves and surfaces

Question. Given space ${X}$ and a multicurve ${\gamma}$ in ${X}$ which is null-homologous. Does ${\gamma}$ bound an essential surface in ${X}$? Essential means ${\pi_1}$-injective.

Interesting cases are

• Surfaces.
• 3-manifolds,
• Shimura varieties.

1. Surface case

Theorem 1 (Calegari) In a higher genus surface, every null-homologous multicurve virtually bounds an immersed surface.

Virtually means that some multiple ${n\gamma}$ does.

Calegari does it by hand, by homologing to simpler multicurves, with less intersections. The figure eight curve in the plane does not bound any immersed surface, but its double does.

2. Case of hyperbolic 3-manifolds

Say a surface ${S}$ immersed in a 3-manifold ${X}$ is panted if a pants decomposition is provided such that the restriction of the immersion to each pant is essential.

Define the panted cobordism group ${\Omega(X)}$ as the set of formal sums of multicurves, modulo cobounding a panted surface.

Theorem 2 (Liu-Markovic) If ${X}$ is a 3-manifold, there is an isomorphism

$\displaystyle \begin{array}{rcl} \Omega(X)\simeq H_1(SO(X),{\mathbb Z}). \end{array}$

where ${SO(X)}$ is the special orthonormal frame bundle of ${X}$.

Using the homotopy exact sequence of the fibration ${SO(X)\rightarrow X}$, get

Corollary 3 There exists a nontrivial ${{\mathbb Z}/2}$-valued invariant ${\sigma}$ on null-homologous multicurves, which is additive and vanishes on boundaries of panted surfaces.

The idea is that cutting a surface along a pants decomposition and regluing it in a different way does not change homology. In this way, one ultimately gets an essential surface.

The same technology yields the following

Theorem 4 (Liu-Markovic) If ${X}$ is a hyperbolic 3-manifold, every rational second homology class has a positive integral multiple represented by an oriented essential quasi-Fuchsian surface.

Indeed, Dehn’s Lemma provide a Haken surface. String it: use a pants decomposition of it, add break the surface.

Sun has continued:

Theorem 5 (Sun) For any finite abelian group ${A}$, every hyperbolic 3-manifold ${X}$ has a finite cover of which the first integral homology has ${A}$ as a direct summand.

Theorem 6 (Sun) For any closed orientable 3-manifold ${N}$, every hyperbolic 3-manifold ${X}$ has a finite cover that 2-dominates ${N}$, meaning that it has degree 2 map to ${N}$.

Degree 2 has been improved to 1 recently. The number 2 arose from our ${{\mathbb Z}/2}$ obstruction above.

3. Case of Shimura varieties

Let ${G}$ be a product of copies of ${PSl(2,{\mathbb R})}$ and ${\Gamma}$ an irreducible lattice in ${G}$. Every example arises from a totally real number field ${K}$, a quaternion algebra over ${K}$ and a maximal order in it.

We show that there exists a surface subgroup ${F\subset\Gamma}$ whose projections to factors are Fuchsian.

Corollary 7 There exists a nontrivial ${{\mathbb Z}/2}$-valued invariant ${\sigma}$ on null-homologous multicurves, which is additive and vanishes on boundaries of essential surfaces.

Theorem 8 (Kahn-Markovic) Let ${K}$ be a totally real number field ${K}$, ${A}$ a quaternion algebra over ${K}$ such that ${A\otimes_K {\mathbb R}\not=M(2,{\mathbb R})}$. Then there exists a Riemann surface ${S}$ whose trace field and quaternion algebra are equal to ${K}$ and ${A}$. Furthermore, group elements of the surface groups are integers in ${A}$.