I looked at a **quadratic cohomology** example. For theoretical backgroun, see the ArXiv paper “Fusion inequality for quadratic cohomology”. It is the case when U is a union of two disjoint smallest open sets in a 2-sphere for which I take the **Icosahedron**, one of the **Platonic solids** and a **2-sphere**, meaning a 2-manifold which becomes contractible if one of the points are taken away. A 2-manifold is a complex for which every unit sphere is a 1-sphere. The Icosahedron is a 2-dimensional simplicial complex G a set of sets with 62 elements. 12 vertices, 30 edges and 20 triangles. With the code posted last week’s blog or on the ArXiv (download the LaTeX source), we can get `G=Whitney[PolyhedronData["Icosahedron","Skeleton"]]`

. Now add the procedure for getting the star of an element in G. It is obtained with the procedure `OpenStar[G_,x_]:=Select[G,SubsetQ[#,x] &]`

. In the example of the code, I took now the union of two open sets `U=Union[OpenStar[G,{1}],OpenStar[G,{2}]]`

and the subsimplicial complex `K=Complement[G,U]`

. This gives for the linear cohomology (simplicial cohomology) the data in the first table below. The Betti data in the first column are topological (even homotopy invariants), while the f-vector in the second column of course depends and just tells how many elements there are from each dimension. The f-vector of the icosahedron G for example is f(G)=(12,30,20), numbers which Rene Descartes already noted (in his secret notebook) to super sum to 2.

[Side remark: Times have changed. At the time of Rene Descartes, mathematicians were paraonid to write down discoveries unencrypted as they feared that their discoveries would be stolen (see the book Aczel about this). Nowadays, the dangers is that discoveries are ignored. Wu characteristic and multi-linear valuations illustrate this. I think that my paper from 2016 about this was after 1970 pretty much the first one, picking it up again. In my opinion, quadratic and higher order characteristics as well as their cohomologies **are as important as Euler characteristic**. It can be even more interesting than linear valuations. It not only generalizes the subject of linear valuations, it also brings in topology. Linear cohomology is a story for homotopy. Quadratic cohomology is a story for topology (a different category). (For example, Quadratic cohomology can distinguish the cylinder from the Moebius strip, two objects which are homotopic but not homeomorphic) Similarly relevant to the work discussed here is the topic of f**inite topologies** which was initiated by Alexandroff in the 1930ies. As interesting finite topological spaces are non-Hausdorff, there had been a decision at one point, (Hausdorff was the mastermind of this coup) to ignore non-Hausdorff cases more or less: of course, the committee picture is only an allegory but it seems that it was FelixHausdorff convinced Heinz Hopf and Pavel Alexandroff to focus in their famous topology book to Hausdorff spaces. The common practice to do a geometric realization of a simplicial complex in some Euclidean space (betraying the subject from the point of view of combinatorial pride), is very misleading as the geometric realization is Hausdorff. But we are in the devils den: first of all, we do not know whether strong enough axiom systems (capable of dealing with real numbers) are consistent or not and even know that we never can prove that we have consistency (both are results of Goedel). Second, the topology of Euclidean space is treacherous. Topological manifolds can be tamed by looking at smooth or piecewise linear manifolds but topological manifolds lead to crazy situations like that the double suspension theorem telling that the double suspension of a homology sphere is always a sphere, while in a combinatorial, finite setting this obviously is not true. Topological manifolds allow for crazy homemorphisms which are not possible in a combinatorial or piecewise linear setting. I myself avoid PL topology from a computer science point of view and prefer to work with finite sets. It would be crazy to implement objects like manifolds using piecewise linear transformations and concrete point implementations in an ambient Euclidean space. It is not only computationally much harder, it is also unnatural as any situation where we need an ambient space to compute properties which are actually intrinsic. Computing the curvature of a surface using an embedding in an ambient space is artificial because we know that it is an intrinsic notion by Gauss’s theorema Egregium. ]

The Betti vectors are the same for all 2-spheres and situations U, where the complement K of U in G is a closed annulus, a manifold with boundary.

```
Case Betti F-vector Euler
----------------------------------------------------
U {0, 0, 2} {2, 10, 10} 2
K {1, 1, 0} {10, 20, 10} 0
G {1, 0, 1} {12, 30, 20} 2
-----------------------------------------------------
Compare {0, 1, 1} {0, 0, 0} 0
```

I should maybe have taken the set `U={{1,3,9},{2,4,12}}`

(because this is the level set of a function and which is a union of two facets. But for linear and quadratic cohomology, the Betti vectors would have been the same. We essentially deal with a zero dimensional sphere modeled as a delta set in a 2-manifold. Open sets U are not simplicial complexes unless U={} or U=G, but still delta sets. So, here is the table which was written down on the blackboard:

```
Case Betti F-vector Wu
---------------------------------------------------------
U {0, 0, 0, 0, 2} {2, 20, 70, 100, 50} 2
K {0, 0, 1, 1, 0} {10, 80, 200, 180, 50} 0
UK {0, 0, 0, 0, 0} {0, 10, 60, 100, 50} 0
KU {0, 0, 0, 0, 0} {0, 10, 60, 100, 50} 0
UU {0, 0, 0, 0, 0} {0, 0, 0, 0, 0} 0
G {0, 0, 1, 0, 1} {12, 120, 390, 480, 200} 2
---------------------------------------------------------
Compare {0, 0, 0, 1, 1} {0, 0, 0, 0, 0} 0
```

If we would have gone with `U={{1,3,9},{2,4,12}}`

, then the table would have been

```
Case Betti F-vector Wu
---------------------------------------------------------
U {0, 0, 0, 0, 2} {0, 0, 0, 0, 2} 2
K {0, 0, 1, 1, 0} {12, 120, 378, 432, 162} 0
UK {0, 0, 0, 0, 0} {0, 0, 6, 24, 18} 0
KU {0, 0, 0, 0, 0} {0, 0, 6, 24, 18} 0
UU {0, 0, 0, 0, 0} {0, 0, 0, 0, 0} 0
G {0, 0, 1, 0, 1} {12, 120, 390, 480, 200} 2
---------------------------------------------------------
Compare {0, 0, 0, 1, 1} {0, 0, 0, 0, 0} 0
```

In the code, I compared the situation with `U=Union[OpenStar[G,{1}],OpenStar[G,{4}]]`

, which is a situation, where U is still the union of two open sets but where `K=Complement[G,U]`

is no more a manifold with boundary. It is still homotopic to a circle but it is an annulus in which part of the annulus has degenerated to a one-dimensional “neck”. This is topologically no more equivalent. It is no more a manfold with boundary. The boundary of the simplicial complex K is a figure 8 object and not the union of two circles. I drew both examples onto the board. The linear simplicial cohomology is the same but the quadratic cohomology now gives different Betti vectors. Here is the second table you see on the board

```
Case Betti F-vector Wu
---------------------------------------------------------
U {0, 0, 0, 0, 2} {2, 20, 70, 100, 50} 2
K {0, 0, 2, 0, 0} {10, 80, 204, 196, 64} 2
UK {0, 0, 0, 2, 0} {0, 10, 56, 84, 36} -2
KU {0, 0, 0, 2, 0} {0, 10, 56, 84, 36} -2
UU {0, 0, 0, 0, 2} {0, 0, 4, 16, 14} 2
G {0, 0, 1, 0, 1} {12, 120, 390, 480, 200} 2
---------------------------------------------------------
Compare {0, 0, 1, 4, 3} {0, 0, 0, 0, 0} 0
```

One can see come to live three additional delta sets UK,U,UU. For UU for example, we look at all pairs (x,y) of elements x,y in U x U, such that is in K. There are 34 elements of this type. In the code, one can list the elements with `WuComplex[U,U,"Open"]`

. It contains for example the pair of edges x={4,6} and y={1,6} or the pair of triangles x={1,5,6}, y={4,5,6}. With `Betti[U,U,"Open"]`

the Betti vector of this situation is computed. By accident, it is the same than `Betti[U,U,"Closed"]`

which is the intrinsic quadratic cohomology of U and which in the paper I denote with b(U). In the