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:
Here is the code for the computation (the proof is the verification that
the code does the right thing):
(* Interaction Cohomology, 3/18/2018 *) (* http://www.math.harvard.edu/~knill/graphgeometry/papers/wu2.pdf *) Coho2[G_,H_]:=Module[{},n=Length[G];m=Length[H];len[x_]:=Total[Map[Length,x]];U={}; Do[If[Length[Intersection[G[[i]],H[[j]]]]>0,U=Append[U,{G[[i]],H[[j]]}]],{i,n},{j,m}]; U=Sort[U,len[#1] < len[#2] & ];u=Length[U];l=Map[len,U]; w=Union[l]; b=Prepend[Table[Max[Flatten[Position[l,w[[k]]]]],{k,Length[w]}],0]; h=Length[b]-1; deriv1[{x_,y_}]:=Table[{Sort[Delete[x,k]],y},{k,Length[x]}]; deriv2[{x_,y_}]:=Table[{x,Sort[Delete[y,k]]},{k,Length[y]}]; d1=Table[0,{u},{u}]; Do[v=deriv1[U[[m]]]; If[Length[v]>0, Do[r=Position[U,v[[k]]]; If[r!={},d1[[m,r[[1,1]]]]=(-1)^k],{k,Length[v]}]],{m,u}]; d2=Table[0,{u},{u}]; Do[v=deriv2[U[[m]]]; If[Length[v]>0, Do[r=Position[U,v[[k]]]; If[r!={},d2[[m,r[[1,1]]]]=(-1)^(Length[U[[m,1]]]+k)], {k,Length[v]}]],{m,u}]; d=d1+d2; Dirac=d+Transpose[d]; L=Dirac.Dirac; Map[NullSpace, Table[Table[L[[b[[k]]+i,b[[k]]+j]],{i,b[[k+1]]-b[[k]]},{j,b[[k+1]]-b[[k]]}],{k,h}]]]; Betti2[G_,H_]:=Map[Length,Coho2[G,H]];Coho2[G_]:=Coho2[G,G]; Betti2[G_]:=Betti2[G,G]; Generate[A_]:=Sort[Map[Sort,Delete[Union[Flatten[Map[Subsets,A],1]],1]]]; duncehat=Generate[{{1,2,5},{1,2,7},{1,2,13},{1,4,8},{1,4,10}, {1,4,11},{1,5,6},{1,6,7},{1,8,9},{1,9,10},{1,11,12},{1,12,13}, {2,5,3},{2,7,3},{2,13,3},{4,8,3},{4,10,3},{4,11,3},{5,3,16}, {5,6,17},{5,16,17},{6,7,17},{7,3,14},{7,14,17},{8,3,14},{8,9,17}, {8,14,17},{9,10,17},{10,3,15},{10,15,17},{11,3,16},{11,12,17}, {11,16,17},{12,13,17},{13,3,15},{13,15,17}}]; Betti2[duncehat]
The dunce hat is interesting because of the open Zeeman conjecture which can nowadays be restated as the claim that for any 2-complex G which is homotopic to a point, there is an interval I such that some Barycentric subdivision of G x I is contractible. (See Corollary 3.5 in Adiprasito-Benedetti 2012). The Zeeman conjecture implies the Perelman theorem (Poincaré conjecture) but unlike the later, the Zeeman conjecture is still open.
Christopher Zeeman passed away just 2 years ago An obituary.
Definition: A simplicial complex $G$ is called contractible, if there exists $x \in G$ such that both the unit sphere $S(x)$ as well as $G \setminus x$ are both contractible. This is a recursive definition. | Definition: Given a locally injective function $f$ on G, a simplex $x \in G$ is called a critical point of $f$ if the stable sphere $S^-(x)=\{ y \in S(x) | f(y) \le f(x) \}$ is not contractible. |
[A) Note that the term “contractible” is used here in the sense what usually is denoted “collapsible”. I like to use the more clear “homotopic to a point” instead of what is often called “collapsible” it might be better not to distinguish between “contractible” and “collapsible” to avoid confusion. Historically, the recursive notion of collapsible in the sense of Whitehead was to the existence of free face, a cell for which the boundary is contractible. The literature on “collapsible” and “contractible” is not uniform. So, again, the Dunce hat is an example of a complex which is homotopic to a point but which is not contractible (collapsible). We could refine the notion of critical point and ask that the stable sphere is not homotopic to the identity but it is not worth adding that complication. Basic definitions like “critical points” need to have a simple definition! So, for us, if a stable sphere of a vertex wold be a dunce hat, then this is considered a critical point. Somehow, our computation that the quadratic interaction cohomology is not trivial justifies this. We can easily build a complex and a function for which the dunce hat appears as a stable sphere: just take a cone extension of the dunce hat and let $f$ have a local maximum at the new vertex. We would call this a critical point.
B) A simplicial complex traditionally is called contractible if there exists a “free face” (a face contained exactly in one other face) such that after removing that face, the remaining complex is contractible. One can see that these definitions are equivalent. The definition sed here origins from Chen-Yau-Yeh and is more practical. One can immediately see what “critical points are” and Poincare-Hopf is almost trivial. Historically the later change of definition appears innocent, but it is important as it leads to a different intuition. To use graphs rather than simplicial complexes was done by Ivashchenko (Evako) who used morphed the Whitehead definition to something more usable. It was the article of Chen-Yau-Yeh “Graph homotopy and Graham homotopy” from 2001 which simplified the Evako homotopy to a more elegant notion, somehow getting rid of the additional Whitehead “free face” part. The use of graphs rather then simplicial complexes is also implicit in Forman’s work as Forman realized the importance to look at the Barycentric refinement (the simplices) as points. ]
By the way, the original paper of Zeeman mentions: (citation) “In general terms the difficulty is one of passing from finite structures (such as complexes) to ordered finite structures (such as handlebodies).” One would rephrase this today as the transition from “finite abstract simplicial complexes” to “discrete CW complexes”. While using CW complexes makes often sense, like in proofs, there is a prize to pay when passing to a construct, in which the order matters.
In 2012, I worked with Frank Josellis (a specialist on Lusternik-Schnirelmann) on a discrete version. Unfortunately, the language of graph theory was not absorbed that nicely. A paper of Aaronson and Scoville from 2013, which uses the language of simplicial complexes had more luck.
The Josellis-Knill paper goes further than the Aaronson-Scoville paper. Things are related as the Barycentric refinement of a simplicial complex is a graph, the results immediately go over to simplicial complexes. Lusternik-Schnirelmann theory is a cousin theory to Morse theory which was developed at about the same time. Both have relations with cohomology. In the case of Morse theory it is the Morse inequalities, in the case of Lusternik-Schnirelmann, it is the inequality between cup length and category.
We need a few definition to state our theorem: let $tcat(G)$ denote the topological Lusternik-Schnirelmann category of a complex. It is the minimal number of in $G$ contractible complexes covering $G$. Let $cat(G)$ be the minimal $tcat(H)$ among all complexes homotopic to $G$.We call this the Lusternik-Schnirelmann category. While $tcat(G)$ is not a homotopy invariant, $cat(H)$ is a homotopy invariant. For the dunce hat $G$ for example, $tcat(G)=2$ but $cat(G)=1$. Let $crit(G)$ denote the minimal number critical points which a locally injective function can have on $G$. A critical point of $f$ is a point for which the stable sphere $S^-_f(x)$ is not contractible. Note that “in $G$ contractible” is not the same than contractible. The dunce hat is contractible but not contractible in itself. Finally, let $cup(G)$ denote the cup length of $G$. As in the continuum, one can form a product $H^i(G) \times H^j(G) \to H^{i+j}(G)$ and make the direct sum of cohomology groups to a graded, non-commutative cohomology ring. The cup length is the largest $k+1$ such that there exist form elements $f_1, \dots, f_k$ in the ring for which $f_1 \cup f_2 \cup \cdots \cup f_k$ is not zero. (the elements have to be p-forms with positive p). |
In other words (in the terminology used above), we can say that a complex is contractible, if $tcat(G)=1$ and homotopic to a point if $cat(G)=1$. We can say that if a complex is cohomologically trivial then $cup(G)=1$ (there is then no p-form with p>0 which is non-trivial). There are smooth 4-manifolds with boundary which are homotopic to a point but not diffeomorphic to the standard 4-ball. These strange balls are called Mazur manifolds, named after Barry Mazur, who constructed them in 1961. The boundary of these Mazur manifolds are homology spheres.
We can characterize a complex as a sphere if $crit(G)=2$. This is the Reeb picture of a sphere. For a sphere we have $cup(G) = cat(G) = crit(G)=2$. For a homology sphere, $cup(G)=2$ and $crit(G)=4$. The Mazur manifolds are examples where cup(G)=1=cat(G)=1 but where crit(G)=3 (there can not be a single minimum and since two critical points are excluded by Reeb). The case $crit(G)=3$ is interesting in the manifold case (which does not apply for the Dunce hat, which is not a manifold). One gets Eells-Kuiper manifolds. They are cohomologically equivalent to the real projective plane, the complex projective plane, the quaternionic projective plane or the Cayley projective plane. It provokes an association with the classification of division algebras (R,C,H,O), which by Hurwitz theorem form a complete list of all division algebras.
Theorem: Josellis-Knill 2012: For any finite abstract simplicial complex, $cup(G) \leq cat(G) \leq tcat(G) \leq crit(G)$. |
Why is this interesting? The cup length is cohomological and so “algebraic”, the category is a homotopy notion and so more “topological” and crit(G) is a Morse theoretical and so an “analytic notion”. The lower bound, the cohomological cup length can be established when knowing the cohomology ring of a complex. The upper bound can be given by providing an explicit function with a given number of critical points.
[In the November 5, 2012 preprint we used only cat(G), which is not a homotopy invariant. This was updated in the November 13, 2012 version. It can be a matter of taste whether one really wants to insist on homotopy invariance. Interaction cohomology produces invariants which are not homotopy invariants.At present, I believe it is better to have a simple result, not trying to optimize.]
Examples::
1) We have $cup(G)=2$ for the octahedron graph as the cohomology is $(1,0,1)$. We can not find $f_1,f_2$ such that $f_1 \cup f_2$ is not zero. Bt we obviously have a 2-form which is non-zero. So, cup(G)=2. We can cover $G$ with two contractible sets. Therefore $cat(G)=2$. At last, there are functions with exactly two critical points so that also $crit(G)=2$.
2) For the 2-torus, we have the cohomology $(1,2,1)$ and $cup(G)=3$ because there are two 1-forms f,g with $f \cup g$ being non-trivial. We have $cat(G)=tcat(G)=crit(G)=3$. (It is not possible to realize 3 critical points with a Morse function as the Poincare-Hopf index is 1 or -1 in that case but there is an example with a critical point of Poincare-Hopf index 2.)
3) For the dunce hat we have we have $cup(G)=1$ (as it is cohomologically equivalent to a point), then $tcat(G)=2$ because it is not contractible within itself. But we have $cat(G)=1$. We have crit(G)=3, but have to note that the dunce hat is not a manifold. In the discrete, it is not a 2-complex: there are unit spheres which are not spheres.
In the literature on Lusternik-Schnirelman, a variety of different definitions appear. Some of them are one larger than or one less than what was given here. We used definitions, such that in “nice situations”, the cup length is equal to the category and is equal to the minimal number of critical points.
What is important however is that the three notions are defined for every simplicial complex and that the inqualities hold unconditionally. The paper with Josellis will certainly be revised at some time with just the definitions, the theorem and some examples which can be done in 3-4 pages but the main difficulty in the matter is obviously just to get the right definitions. The discrete situation is not much different from the continuous.
At the moment, I believe should again go back and not insist on homotopy invariance of category and just state the simpler theorem:
$cup(G) \leq cat(G() \leq cri(G)$. |
and not worry that both cat and cri are not homotopy invariant notions. One of the nice things one can do in the discrete is count. It would be interesting to know the statistics of category. What is the average Lusternik-Schnirelman category among all simplicial complexes with n elements? How frequent is it that the cup length is different from the category? How frequent is it that the minimal number of critical points is larger?