quadratic interaction Betti vector of the dunce hat

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1. (* Interaction Cohomology, 3/18/2018                                *)
2. (* http://www.math.harvard.edu/~knill/graphgeometry/papers/wu2.pdf  *)
3. Coho2[G_,H_]:=Module[{},n=Length[G];m=Length[H];len[x_]:=Total[Map[Length,x]];U={};
4.   Do[If[Length[Intersection[G[[i]],H[[j]]]]>0,U=Append[U,{G[[i]],H[[j]]}]],{i,n},{j,m}];
5.   U=Sort[U,len[#1] < len[#2] & ];u=Length[U];l=Map[len,U]; w=Union[l];
6.   b=Prepend[Table[Max[Flatten[Position[l,w[[k]]]]],{k,Length[w]}],0]; h=Length[b]-1;
7.   deriv1[{x_,y_}]:=Table[{Sort[Delete[x,k]],y},{k,Length[x]}];
8.   deriv2[{x_,y_}]:=Table[{x,Sort[Delete[y,k]]},{k,Length[y]}];
9.   d1=Table[0,{u},{u}]; Do[v=deriv1[U[[m]]]; If[Length[v]>0,
10.     Do[r=Position[U,v[[k]]]; If[r!={},d1[[m,r[[1,1]]]]=(-1)^k],{k,Length[v]}]],{m,u}];
11.   d2=Table[0,{u},{u}]; Do[v=deriv2[U[[m]]]; If[Length[v]>0,
12.     Do[r=Position[U,v[[k]]]; If[r!={},d2[[m,r[[1,1]]]]=(-1)^(Length[U[[m,1]]]+k)],
13.     {k,Length[v]}]],{m,u}]; d=d1+d2; Dirac=d+Transpose[d]; L=Dirac.Dirac; Map[NullSpace,
14.   Table[Table[L[[b[[k]]+i,b[[k]]+j]],{i,b[[k+1]]-b[[k]]},{j,b[[k+1]]-b[[k]]}],{k,h}]]];
15. Betti2[G_,H_]:=Map[Length,Coho2[G,H]];Coho2[G_]:=Coho2[G,G]; Betti2[G_]:=Betti2[G,G];
16. Generate[A_]:=Sort[Map[Sort,Delete[Union[Flatten[Map[Subsets,A],1]],1]]];
17. duncehat=Generate[{{1,2,5},{1,2,7},{1,2,13},{1,4,8},{1,4,10},
18. {1,4,11},{1,5,6},{1,6,7},{1,8,9},{1,9,10},{1,11,12},{1,12,13},
19. {2,5,3},{2,7,3},{2,13,3},{4,8,3},{4,10,3},{4,11,3},{5,3,16},
20. {5,6,17},{5,16,17},{6,7,17},{7,3,14},{7,14,17},{8,3,14},{8,9,17},
21. {8,14,17},{9,10,17},{10,3,15},{10,15,17},{11,3,16},{11,12,17},
22. {11,16,17},{12,13,17},{13,3,15},{13,15,17}}]; Betti2[duncehat]

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The Cohomology for Wu characteristics
The dunce hat and Lusternik-Schnirelmann  By oliverknill
The dunce hat of Zeeman was introduced in 1964 as an example of a topological space which is homotopic to a point but which is not contractible (I use here contractible as a synonym to collapsible, for reasons seen below). Also from 1964 is the house with two rooms of Bing. As the dunce hat is homotopic to a point, its simplicial cohomology is trivial. The Betti vector is (1,0,0). The main point, I want to make here first that quadratic interaction cohomology is interesting for the dunce hat. We have:

Computation: The quadratic interaction Betti vector of the dunce hat is $(0,0,0,1,2)$. It leads to the Wu characteristic 1. So, we can cohomologically distinguish the dunce hat from a topological disc for example.

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dunce hat