Bruno Martelli's research activity

I work in geometric topology. My main research interest concerns the construction and study of 3-manifolds and 4-manifolds via combinatorial and geometric tools like spines, polyhedra, and Dehn surgery. I am also interested in hyperbolic geometry and quantum topology.

Complexity of 3-manifolds:

As defined by Matveev, the complexity of a 3-manifold is the minimum number of vertices of a simple spine, and equals the minimum number of tetrahedra in a triangulation in the most interesting cases. With various collaborators and one computer, we have produced tables of:

  • closed orientable 3-manifolds up to complexity 9 [2] and 10 [12], with Petronio;
  • closed non-orientable 3-manifolds up to complexity 6 [5] and 7 [9], with Amendola;
  • compact hyperbolic manifolds with geodesic boundary and possibly some cusps up to complexity 4 [7], with Frigerio and Petronio;
  • hyperbolic graphs in 3-manifolds up to complexity 5 [16], with Heard, Hodgson, and Petronio.
The manifolds are listed in these tables (from Carlo's web page). Some related census exist in the literature, and all overlaps are luckily coherent! You can find:
  • cusped hyperbolic 3-manifolds up to complexity 7, from SnapPea census by Callahan, Hildebrand, and Weeks;
  • closed 3-manifolds of complexity up to 11, from Burton's Regina;
  • closed orientable 3-manifolds of complexity up to 12, from Matveev's atlas of 3-manifolds.
Some related papers I wrote contain:
  • a conjectural formula for the complexity of all Seifert manifolds [8], with Petronio;
  • a survey [12];
  • a recent study of the stable complexity, i.e. the complexity stabilized under finite covers [20], with Francaviglia and Frigerio.
When you list thousands of manifolds, you inevitably tumble on some funny families:
  • those having an ideal triangulation with only one edge [4], with Frigerio and Petronio;
  • some knots in handlebodies with curious Dehn surgeries [6], with Frigerio and Petronio;
  • pick any triangulation and replace every tetrahedron with an ideal regular hyperbolic octahedron [14], with Costantino, Frigerio, and Petronio;

Complexity of 4-manifolds:

I would like to define a reasonable complexity on closed orientable 4-manifolds. I have tried essentially two approaches:

  • use 3-dimensional spines [17];
  • use 2-dimensional polyhedra (called shadows by Turaev) [18].
I have spent a lot of time recently on each. The "shadow" approach is combinatorially more treatable since it is very much connected to 3-dimensional topology and Dehn surgery. The 4-manifolds having complexity zero look very much like Waldhausen's graph manifolds. I am curious to see what happens with complexity 1, 2, ...

Dehn surgery:

A Dehn surgery is exceptional when the resulting 3-manifold is not hyperbolic. A lot of effort has been devoted in the last 30 years to understanding exceptional surgeries: see this recent survey of Cameron Gordon. An extremely ambituous question would be the following: can we understand all exceptional surgeries on all knots/links in the 3-sphere?

There are infinitely many exceptional fillings on a multicomponent link, but it is possible to describe all of them with a finite amount of data. Such data form sometimes a surprisingly short list, which can be prouduced via computer in a reasonably short amunt of time. Consider for instance the following chain links

These are conjecturally the links with i = 1, ..., 5 components having smallest hyperbolic volume. Using a python code on SnapPy it is possible to classify all exceptional surgeries on such links in few minutes, and the amount of data necessary to describe them is roughly constant for i = 1, 2, ..., 5: on each link, less than 10 exceptional surgeries are enough to "generate" (via symmetries) all the exceptional surgeries. See [22], with Petronio and Roukema.

The python code is available on this page and can be used on any link. See the detailed instructions there.

Normal surfaces:

Normal and (octagonal) almost normal surfaces generalize to k-normal surfaces, belonging the cases k=0 and 1 respectively. With Evgeny Fominykh we gave a short proof that a minimal triangulation of an irreducible 3-manifold does not contain any k-normal sphere (with few exceptions) in [15].

Kirby moves:

The short paper [19] answers a nice question on Mathoverflow about Kirby calculus. I show that there is a finite collection of local moves that connected any two surgery presentations of the same 3-manifold via framed links in the three-sphere.

Quantum invariants:

I have more recently been interested in quantum invariants. In [23] with Costantino we use these mysterious objects to construct an analytic family of representations of the mapping class group defined on the unit disc, that includes the finite representations at the roots of unity. It would be nice to get similar analytic families for other 3-dimensional objects.

In [25] with Carrega we study the relation between quantum invariants, shadows, and ribbon surfaces. We have extended a theorem of Eisermann that connects quantum invariant and ribbon surfaces in the 3-spheres.

Hyperbolic 4-manifolds:

Various hyperbolic manifolds can be constructed by assembling hyperbolic regular polytopes. In [21] with Kolpakov we use the ideal hyperbolic 24-cell to build hyperbolic four-manifolds with an arbitrary number of cusps, and in [24] with Kolpakov and Tschantz we use the 120-cell to build some hyperbolic four-manifolds with connected geodesic boundary of controlled volume. In [27] I prove that every finite-volume hyperbolic 3-manifold that decomposes into right-angled regular polyhedra is geodesically contained in a hyperbolic 4-manifold having controlled volume.

The paper [29] is a survey on hyperbolic four-manifolds. In [30] with Riolo we define a deformation relating two non-commensurable hyperbolic four-manifolds through cone manifolds with cone singularity an immersed surface. This family may be interpreted as a hyperbolic Dehn filling.

Tropical geometry:

I have been interested in tropical geometry for some time. In [26] with Golla we study the topological notion of decomposing a four-manifold into pair-of-pants that arises naturally from this area.

Spines of minimal area

In [28], with Novaga, Pluda, and Riolo, we raise the question whether every closed riemannian manifold has a spine of minimal area (that is, codimension one Hausdorff measure). We answer it affirmatively in dimension 2 and study the spines of minimal lengths on constant curvature surfaces. We introduce the spine systole, a proper function on moduli spaces.

some paintings of my mother

  1. Minimal spines and geometric decompositions of closed 3-manifolds
    Proceedings of the conference "Low-dimensional topology and combinatorial group theory (Chelyabinsk 1999)'', 215-226,
    Inst. of Math. of Nat. Acad. Sci. of Ukraine, Kiev

  2. (with C. Petronio) 3-manifolds having complexity at most 9
    Experimental Math. 10 (2001), 207-237

  3. (with C. Petronio) A new decomposition theorem for 3-manifolds
    Illinois J. Math. 46 (2002), 755-780

  4. (with R. Frigerio and C. Petronio) Complexity and Heegaard genus of an infinite class of compact 3-manifolds
    Pacific J. Math. 210 (2003), 283-297

  5. (with G. Amendola) Non-orientable 3-manifolds of small complexity
    Topology Appl. 133 (2003), 157-178

  6. (with R. Frigerio and C. Petronio) Dehn filling of cusped hyperbolic 3-manifolds with geodesic boundary
    J. Diff. Geom. 64 (2003), 425-456

  7. (with R. Frigerio and C. Petronio) Small hyperbolic 3-manifolds with geodesic boundary
    Experimental Math. 13 (2004), 177-190

  8. (with C. Petronio) Complexity of geometric three-manifolds
    Geom. Dedicata 108 (2004), 15-69

  9. (with G. Amendola) Non-orientable 3-manifolds of complexity up to 7
    Topology Appl. 150 (2005), 179-195

  10. Links, two-handles, and four-manifolds
    Int. Math. Res. Not. 58 (2005), 3595-3624

  11. (with C. Petronio) Dehn filling of the `magic' 3-manifold
    Comm. Anal. Geom. 14 (2006), 967-1024

  12. Complexity of 3-manifolds
    "Spaces of Kleinian groups", London Math. Soc. Lec. Notes Ser. 329 (2006), 91-120

  13. (with R. Frigerio) Countable groups are mapping class groups of hyperbolic 3-manifolds
    Math. Res. Lett. 13 (2006), 897-910

  14. (with F. Costantino, R. Frigerio, and C. Petronio) Triangulations of 3-manifolds, hyperbolic relative handlebodies, and Dehn filling
    Comm. Math. Helv. 82 (2007), 903-934

  15. (with E. Fominykh) k-Normal surfaces
    J. Diff. Geom. 82 (2009), 101-114

  16. (with D. Heard, C. Hodgson, and C. Petronio) Hyperbolic graphs of small complexity
    Experimental Math. 19 (2010), 211-236

  17. Complexity of PL manifolds
    Algebraic & Geometric Topology 10 (2010), 1107-1164

  18. Four-manifolds with shadow-complexity zero
    Int. Math. Res. Not. 2011 (2011), 1268-1351

  19. A finite set of local moves for Kirby calculus
    J. Knot Theory Ramif. 21 (2012), 1250126

  20. (with S. Francaviglia and R. Frigerio) Stable complexity and simplicial volume of manifolds
    Journal of Topology 5 (2012), 977-1010

  21. (with A. Kolpakov) Hyperbolic four-manifolds with one cusp
    Geom. & Funct. Anal. 23 (2013), 1903-1933

  22. (with F. Costantino) An analytic family of representations for the mapping class group of punctured surfaces
    Geometry & Topology 18 (2014) 1485-1538

  23. (with C. Petronio and F. Roukema) Exceptional Dehn surgery on the minimally twisted five-chain link
    Comm. Anal. Geom. 22 (2014) 689-735

  24. (with A. Kolpakov and S. Tschantz) Some hyperbolic three-manifolds that bound geometrically
    Proc. Amer. Math. Soc. 143 (2015) 4103-4111

  25. (with A. Carrega) Shadows, ribbon surfaces, and quantum invariants
    Quantum Topology 8 (2017) 249-294

  26. (with M. Novaga, A. Pluda, and S. Riolo) Spines of minimal length
    arXiv:1511.02367, to appear in Ann. Sc. Norm. Sup. Pisa Cl. Sci

  27. (with M. Golla) Pair of pants decomposition of 4-manifolds
    arXiv:1503.05839, to appear in Algebraic & Geometric Topology

  28. Hyperbolic three-manifolds that embed geodesically

  29. Hyperbolic four-manifolds
    arXiv:1512.03661, submitted

  30. (with S. Riolo) Hyperbolic Dehn filling in dimension four
    arXiv:1608.08309, submitted
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