Recognizing Manifolds

This is about combinatorial topology in polymake from within Oscar (via Polymake.jl).

Author: Michael Joswig

In [1]:
using Oscar
const pm = Polymake
 -----    -----    -----      -      -----   
|     |  |     |  |     |    | |    |     |  
|     |  |        |         |   |   |     |  
|     |   -----   |        |     |  |-----   
|     |        |  |        |-----|  |   |    
|     |  |     |  |     |  |     |  |    |   
 -----    -----    -----   -     -  -     -  

...combining (and extending) ANTIC, GAP, Polymake and Singular
Version 0.5.1 ... 
 ... which comes with absolutely no warranty whatsoever
Type: '?Oscar' for more information
(c) 2019-2021 by The Oscar Development Team
Out[1]:
Polymake

Casella and Kühnel, Topology 40 (2001), found a 16-vertex triangulation of a K3 surface, and this is the minimum number. While there are many K3 surfaces which are distinct as algebraic varieties, topologically they are all the same.

In [2]:
K3 = pm.topaz.SimplicialComplex(FACETS=[[0,1,2,3,7],[0,1,2,3,11],[0,1,2,7,11],[0,1,3,4,9],[0,1,3,4,13],[0,1,3,5,11],[0,1,3,5,13],[0,1,3,7,9],[0,1,4,6,12],[0,1,4,6,14],[0,1,4,9,11],[0,1,4,11,12],[0,1,4,13,14],[0,1,5,6,8],[0,1,5,6,12],[0,1,5,8,14],[0,1,5,11,12],[0,1,5,13,14],[0,1,6,8,10],[0,1,6,10,15],[0,1,6,14,15],[0,1,7,9,11],[0,1,8,10,14],[0,1,10,14,15],[0,2,3,6,8],[0,2,3,6,10],[0,2,3,7,8],[0,2,3,10,12],[0,2,3,11,12],[0,2,4,5,8],[0,2,4,5,10],[0,2,4,7,8],[0,2,4,7,10],[0,2,5,8,14],[0,2,5,9,10],[0,2,5,9,15],[0,2,5,13,14],[0,2,5,13,15],[0,2,6,8,9],[0,2,6,9,10],[0,2,7,10,12],[0,2,7,11,12],[0,2,8,9,14],[0,2,9,13,14],[0,2,9,13,15],[0,3,4,7,9],[0,3,4,7,13],[0,3,5,6,8],[0,3,5,6,12],[0,3,5,7,8],[0,3,5,7,13],[0,3,5,11,12],[0,3,6,10,15],[0,3,6,12,14],[0,3,6,14,15],[0,3,10,12,15],[0,3,12,14,15],[0,4,5,7,8],[0,4,5,7,13],[0,4,5,9,10],[0,4,5,9,15],[0,4,5,13,15],[0,4,6,11,12],[0,4,6,11,14],[0,4,7,9,10],[0,4,9,11,15],[0,4,11,13,14],[0,4,11,13,15],[0,6,7,11,12],[0,6,7,11,14],[0,6,7,12,14],[0,6,8,9,13],[0,6,8,10,13],[0,6,9,10,13],[0,7,9,10,12],[0,7,9,11,15],[0,7,9,12,15],[0,7,11,14,15],[0,7,12,14,15],[0,8,9,13,14],[0,8,10,11,14],[0,8,10,11,15],[0,8,10,12,13],[0,8,10,12,15],[0,8,11,13,14],[0,8,11,13,15],[0,8,12,13,15],[0,9,10,12,13],[0,9,12,13,15],[0,10,11,14,15],[1,2,3,6,7],[1,2,3,6,11],[1,2,4,5,8],[1,2,4,5,12],[1,2,4,8,15],[1,2,4,12,15],[1,2,5,8,14],[1,2,5,12,14],[1,2,6,7,13],[1,2,6,11,13],[1,2,7,9,13],[1,2,7,9,14],[1,2,7,11,14],[1,2,8,10,13],[1,2,8,10,14],[1,2,8,12,13],[1,2,8,12,15],[1,2,9,12,13],[1,2,9,12,14],[1,2,10,11,13],[1,2,10,11,14],[1,3,4,6,9],[1,3,4,6,13],[1,3,5,11,14],[1,3,5,13,14],[1,3,6,7,15],[1,3,6,9,15],[1,3,6,11,13],[1,3,7,8,12],[1,3,7,8,15],[1,3,7,9,12],[1,3,8,9,12],[1,3,8,9,15],[1,3,10,11,13],[1,3,10,11,14],[1,3,10,13,14],[1,4,5,7,8],[1,4,5,7,15],[1,4,5,9,10],[1,4,5,9,15],[1,4,5,10,12],[1,4,6,9,10],[1,4,6,10,12],[1,4,6,13,14],[1,4,7,8,15],[1,4,9,11,15],[1,4,11,12,15],[1,5,6,8,10],[1,5,6,10,12],[1,5,7,8,10],[1,5,7,10,15],[1,5,9,10,15],[1,5,11,12,14],[1,6,7,13,15],[1,6,9,10,15],[1,6,13,14,15],[1,7,8,10,13],[1,7,8,12,13],[1,7,9,11,14],[1,7,9,12,13],[1,7,10,13,15],[1,8,9,11,12],[1,8,9,11,15],[1,8,11,12,15],[1,9,11,12,14],[1,10,13,14,15],[2,3,4,5,10],[2,3,4,5,14],[2,3,4,10,15],[2,3,4,14,15],[2,3,5,9,10],[2,3,5,9,11],[2,3,5,11,15],[2,3,5,13,14],[2,3,5,13,15],[2,3,6,7,8],[2,3,6,9,10],[2,3,6,9,11],[2,3,10,12,15],[2,3,11,12,15],[2,3,13,14,15],[2,4,5,12,14],[2,4,6,12,13],[2,4,6,12,15],[2,4,6,13,14],[2,4,6,14,15],[2,4,7,8,11],[2,4,7,10,11],[2,4,8,10,11],[2,4,8,10,15],[2,4,9,12,13],[2,4,9,12,14],[2,4,9,13,14],[2,5,6,7,9],[2,5,6,7,15],[2,5,6,9,11],[2,5,6,11,15],[2,5,7,9,15],[2,6,7,8,9],[2,6,7,13,15],[2,6,11,12,13],[2,6,11,12,15],[2,6,13,14,15],[2,7,8,9,14],[2,7,8,11,14],[2,7,9,13,15],[2,7,10,11,12],[2,8,10,11,14],[2,8,10,12,13],[2,8,10,12,15],[2,10,11,12,13],[3,4,5,10,14],[3,4,6,9,13],[3,4,7,9,12],[3,4,7,10,11],[3,4,7,10,14],[3,4,7,11,13],[3,4,7,12,14],[3,4,8,9,12],[3,4,8,9,13],[3,4,8,10,11],[3,4,8,10,15],[3,4,8,11,13],[3,4,8,12,14],[3,4,8,14,15],[3,5,6,8,12],[3,5,7,8,12],[3,5,7,12,13],[3,5,9,10,14],[3,5,9,11,14],[3,5,11,12,15],[3,5,12,13,15],[3,6,7,8,15],[3,6,8,12,14],[3,6,8,14,15],[3,6,9,10,15],[3,6,9,11,13],[3,7,10,11,13],[3,7,10,13,14],[3,7,12,13,14],[3,8,9,10,11],[3,8,9,10,15],[3,8,9,11,13],[3,9,10,11,14],[3,12,13,14,15],[4,5,6,7,11],[4,5,6,7,15],[4,5,6,11,15],[4,5,7,11,13],[4,5,10,12,14],[4,5,11,13,15],[4,6,7,8,11],[4,6,7,8,15],[4,6,8,11,14],[4,6,8,14,15],[4,6,9,10,13],[4,6,10,12,13],[4,6,11,12,15],[4,7,9,10,12],[4,7,10,12,14],[4,8,9,12,14],[4,8,9,13,14],[4,8,11,13,14],[4,9,10,12,13],[5,6,7,9,14],[5,6,7,10,11],[5,6,7,10,14],[5,6,8,9,12],[5,6,8,9,13],[5,6,8,10,13],[5,6,9,11,13],[5,6,9,12,14],[5,6,10,11,13],[5,6,10,12,14],[5,7,8,10,13],[5,7,8,12,13],[5,7,9,14,15],[5,7,10,11,13],[5,7,10,14,15],[5,8,9,11,12],[5,8,9,11,13],[5,8,11,12,15],[5,8,11,13,15],[5,8,12,13,15],[5,9,10,14,15],[5,9,11,12,14],[6,7,8,9,14],[6,7,8,11,14],[6,7,10,11,12],[6,7,10,12,14],[6,8,9,12,14],[6,10,11,12,13],[7,9,11,14,15],[7,9,12,13,15],[7,10,13,14,15],[7,12,13,14,15],[8,9,10,11,15],[9,10,11,14,15]])
Out[2]:
type
SimplicialComplex
FACETS
{0 1 2 3 7}
{0 1 2 3 11}
{0 1 2 7 11}
{0 1 3 4 9}
{0 1 3 4 13}
{0 1 3 5 11}
{0 1 3 5 13}
{0 1 3 7 9}
{0 1 4 6 12}
{0 1 4 6 14}
{0 1 4 9 11}
{0 1 4 11 12}
{0 1 4 13 14}
{0 1 5 6 8}
{0 1 5 6 12}
{0 1 5 8 14}
{0 1 5 11 12}
{0 1 5 13 14}
{0 1 6 8 10}
{0 1 6 10 15}
{0 1 6 14 15}
{0 1 7 9 11}
{0 1 8 10 14}
{0 1 10 14 15}
{0 2 3 6 8}
{0 2 3 6 10}
{0 2 3 7 8}
{0 2 3 10 12}
{0 2 3 11 12}
{0 2 4 5 8}
{0 2 4 5 10}
{0 2 4 7 8}
{0 2 4 7 10}
{0 2 5 8 14}
{0 2 5 9 10}
{0 2 5 9 15}
{0 2 5 13 14}
{0 2 5 13 15}
{0 2 6 8 9}
{0 2 6 9 10}
{0 2 7 10 12}
{0 2 7 11 12}
{0 2 8 9 14}
{0 2 9 13 14}
{0 2 9 13 15}
{0 3 4 7 9}
{0 3 4 7 13}
{0 3 5 6 8}
{0 3 5 6 12}
{0 3 5 7 8}
{0 3 5 7 13}
{0 3 5 11 12}
{0 3 6 10 15}
{0 3 6 12 14}
{0 3 6 14 15}
{0 3 10 12 15}
{0 3 12 14 15}
{0 4 5 7 8}
{0 4 5 7 13}
{0 4 5 9 10}
{0 4 5 9 15}
{0 4 5 13 15}
{0 4 6 11 12}
{0 4 6 11 14}
{0 4 7 9 10}
{0 4 9 11 15}
{0 4 11 13 14}
{0 4 11 13 15}
{0 6 7 11 12}
{0 6 7 11 14}
{0 6 7 12 14}
{0 6 8 9 13}
{0 6 8 10 13}
{0 6 9 10 13}
{0 7 9 10 12}
{0 7 9 11 15}
{0 7 9 12 15}
{0 7 11 14 15}
{0 7 12 14 15}
{0 8 9 13 14}
{0 8 10 11 14}
{0 8 10 11 15}
{0 8 10 12 13}
{0 8 10 12 15}
{0 8 11 13 14}
{0 8 11 13 15}
{0 8 12 13 15}
{0 9 10 12 13}
{0 9 12 13 15}
{0 10 11 14 15}
{1 2 3 6 7}
{1 2 3 6 11}
{1 2 4 5 8}
{1 2 4 5 12}
{1 2 4 8 15}
{1 2 4 12 15}
{1 2 5 8 14}
{1 2 5 12 14}
{1 2 6 7 13}
{1 2 6 11 13}
{1 2 7 9 13}
{1 2 7 9 14}
{1 2 7 11 14}
{1 2 8 10 13}
{1 2 8 10 14}
{1 2 8 12 13}
{1 2 8 12 15}
{1 2 9 12 13}
{1 2 9 12 14}
{1 2 10 11 13}
{1 2 10 11 14}
{1 3 4 6 9}
{1 3 4 6 13}
{1 3 5 11 14}
{1 3 5 13 14}
{1 3 6 7 15}
{1 3 6 9 15}
{1 3 6 11 13}
{1 3 7 8 12}
{1 3 7 8 15}
{1 3 7 9 12}
{1 3 8 9 12}
{1 3 8 9 15}
{1 3 10 11 13}
{1 3 10 11 14}
{1 3 10 13 14}
{1 4 5 7 8}
{1 4 5 7 15}
{1 4 5 9 10}
{1 4 5 9 15}
{1 4 5 10 12}
{1 4 6 9 10}
{1 4 6 10 12}
{1 4 6 13 14}
{1 4 7 8 15}
{1 4 9 11 15}
{1 4 11 12 15}
{1 5 6 8 10}
{1 5 6 10 12}
{1 5 7 8 10}
{1 5 7 10 15}
{1 5 9 10 15}
{1 5 11 12 14}
{1 6 7 13 15}
{1 6 9 10 15}
{1 6 13 14 15}
{1 7 8 10 13}
{1 7 8 12 13}
{1 7 9 11 14}
{1 7 9 12 13}
{1 7 10 13 15}
{1 8 9 11 12}
{1 8 9 11 15}
{1 8 11 12 15}
{1 9 11 12 14}
{1 10 13 14 15}
{2 3 4 5 10}
{2 3 4 5 14}
{2 3 4 10 15}
{2 3 4 14 15}
{2 3 5 9 10}
{2 3 5 9 11}
{2 3 5 11 15}
{2 3 5 13 14}
{2 3 5 13 15}
{2 3 6 7 8}
{2 3 6 9 10}
{2 3 6 9 11}
{2 3 10 12 15}
{2 3 11 12 15}
{2 3 13 14 15}
{2 4 5 12 14}
{2 4 6 12 13}
{2 4 6 12 15}
{2 4 6 13 14}
{2 4 6 14 15}
{2 4 7 8 11}
{2 4 7 10 11}
{2 4 8 10 11}
{2 4 8 10 15}
{2 4 9 12 13}
{2 4 9 12 14}
{2 4 9 13 14}
{2 5 6 7 9}
{2 5 6 7 15}
{2 5 6 9 11}
{2 5 6 11 15}
{2 5 7 9 15}
{2 6 7 8 9}
{2 6 7 13 15}
{2 6 11 12 13}
{2 6 11 12 15}
{2 6 13 14 15}
{2 7 8 9 14}
{2 7 8 11 14}
{2 7 9 13 15}
{2 7 10 11 12}
{2 8 10 11 14}
{2 8 10 12 13}
{2 8 10 12 15}
{2 10 11 12 13}
{3 4 5 10 14}
{3 4 6 9 13}
{3 4 7 9 12}
{3 4 7 10 11}
{3 4 7 10 14}
{3 4 7 11 13}
{3 4 7 12 14}
{3 4 8 9 12}
{3 4 8 9 13}
{3 4 8 10 11}
{3 4 8 10 15}
{3 4 8 11 13}
{3 4 8 12 14}
{3 4 8 14 15}
{3 5 6 8 12}
{3 5 7 8 12}
{3 5 7 12 13}
{3 5 9 10 14}
{3 5 9 11 14}
{3 5 11 12 15}
{3 5 12 13 15}
{3 6 7 8 15}
{3 6 8 12 14}
{3 6 8 14 15}
{3 6 9 10 15}
{3 6 9 11 13}
{3 7 10 11 13}
{3 7 10 13 14}
{3 7 12 13 14}
{3 8 9 10 11}
{3 8 9 10 15}
{3 8 9 11 13}
{3 9 10 11 14}
{3 12 13 14 15}
{4 5 6 7 11}
{4 5 6 7 15}
{4 5 6 11 15}
{4 5 7 11 13}
{4 5 10 12 14}
{4 5 11 13 15}
{4 6 7 8 11}
{4 6 7 8 15}
{4 6 8 11 14}
{4 6 8 14 15}
{4 6 9 10 13}
{4 6 10 12 13}
{4 6 11 12 15}
{4 7 9 10 12}
{4 7 10 12 14}
{4 8 9 12 14}
{4 8 9 13 14}
{4 8 11 13 14}
{4 9 10 12 13}
{5 6 7 9 14}
{5 6 7 10 11}
{5 6 7 10 14}
{5 6 8 9 12}
{5 6 8 9 13}
{5 6 8 10 13}
{5 6 9 11 13}
{5 6 9 12 14}
{5 6 10 11 13}
{5 6 10 12 14}
{5 7 8 10 13}
{5 7 8 12 13}
{5 7 9 14 15}
{5 7 10 11 13}
{5 7 10 14 15}
{5 8 9 11 12}
{5 8 9 11 13}
{5 8 11 12 15}
{5 8 11 13 15}
{5 8 12 13 15}
{5 9 10 14 15}
{5 9 11 12 14}
{6 7 8 9 14}
{6 7 8 11 14}
{6 7 10 11 12}
{6 7 10 12 14}
{6 8 9 12 14}
{6 10 11 12 13}
{7 9 11 14 15}
{7 9 12 13 15}
{7 10 13 14 15}
{7 12 13 14 15}
{8 9 10 11 15}
{9 10 11 14 15}

We now give a computational proof that this simplicial complex is indeed a K3 surface. This first appeared in Joswig, Proc. ICMS, Beijing (2002); see also Spreer & Kühnel, Exp. Math. 20 (2011).

In [3]:
@show K3.DIM
@show K3.F_VECTOR
K3.DIM = 4
K3.F_VECTOR = pm::Array<long>
16 120 560 720 288
Out[3]:
pm::Array<long>
16 120 560 720 288

In general, it is hard to check if a space is simply connected or not. However, here it happens that all possible triangles are there. Thus, in particular, the space is simply connected.

In [4]:
f = K3.F_VECTOR
pm.common.binomial(f[1],3) == f[3]
Out[4]:
true

Another tricky thing is to check whether a given space is a manifold. The most recent polymake version employs a heuristic method by Joswig, Lofano, Lutz & Tsuruga, arXiv:1405.3848. The idea is to provide certificates (ve.g., via discrete Morse theory) to show that the link of each face is a sphere (of the proper dimension). Note that sphere recognition is undecidable in general.

In [5]:
K3.MANIFOLD
Out[5]:
true

Now we know that the given space is a simply connected 4-manifold.

The (reduced) cohomology looks like it should (rank 22 in degree 2 and rank 1 in degree 4). In particular, wee see that the only nontrivial contribution comes from products of 2-cocycles.

In [6]:
K3.COHOMOLOGY
Out[6]:
PropertyValue wrapping pm::Array<polymake::topaz::HomologyGroup<pm::Integer>>
({} 0)
({} 0)
({} 22)
({} 0)
({} 1)

That is, the multiplication in the cohomology algebra is determined by an integral quadratic form, the intersection form of the 4-manifold.

In [7]:
K3.INTERSECTION_FORM
Out[7]:
PropertyValue wrapping polymake::topaz::IntersectionForm
0 3 19

So we obtain an even intersection form of signature $(3,19)$. By a celebrated result of Freedman, J. Differential Geom. 17,3 (1982) this characterizes the K3-surface.