** Homology of curves and surfaces **

**Question**. Given space and a multicurve in which is null-homologous. Does bound an essential surface in ? Essential means -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 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 immersed in a 3-manifold 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* as the set of formal sums of multicurves, modulo cobounding a panted surface.

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

where is the special orthonormal frame bundle of .

Using the homotopy exact sequence of the fibration , get

Corollary 3There exists a nontrivial -valued invariant 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 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 , every hyperbolic 3-manifold has a finite cover of which the first integral homology has as a direct summand.

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

Degree 2 has been improved to 1 recently. The number 2 arose from our obstruction above.

**3. Case of Shimura varieties **

Let be a product of copies of and an irreducible lattice in . Every example arises from a totally real number field , a quaternion algebra over and a maximal order in it.

We show that there exists a surface subgroup whose projections to factors are Fuchsian.

Corollary 7There exists a nontrivial -valued invariant on null-homologous multicurves, which is additive and vanishes on boundaries of essential surfaces.

Theorem 8 (Kahn-Markovic)Let be a totally real number field , a quaternion algebra over such that . Then there exists a Riemann surface whose trace field and quaternion algebra are equal to and . Furthermore, group elements of the surface groups are integers in .