Fusion inequality for quadratic cohomology

We prove here that if $K$ is a sub-complex of a finite abstract simplicial complex $G$ and $U$ is the open complement $U=G\setminus K$ [1, 12], there are besides the quadratic cohomology of $G$ five quadratic cohomology groups belonging to the five quadratic Hodge Laplacians $L(X),L(X,Y)$ with $X,Y\in\{U,K\}$. They all satisfy spectral inequalities:

The assumption is that all eigenvalues are ordered in an ascending order and that they are padded left in comparision with the eigenvalues of $G$. This result parallels the linear simplicial cohomology case [13], where $U$ and $K$ can not yet interact and $L$ is one of the Hodge Laplacians $L(K),L(U)$ for simplicial cohomology.

In the linear case, the Betti vectors satisfied the fusion inequality $b(G)\leq b(K)+b(U)$ [10]. This linear fusion inequality had followed from the spectral inequality and the fact that cohomology classes are null-spaces of matrices. Counting gave $f(G)=f(K)+f(U)$ for the $f$-vectors and the Euler-Poincaré formula was $\chi(X)=\sum_{k}(-1)^{k}f_{k}(X)=\sum_{k}(-1)^{k}b_{k}(X)$ for $X\in\{G,K,U\}$, seen directly by heat deformation using the McKean-Singer symmetry, rephrasing that $D_{X}$ is an isomorphism between even and odd parts of image of the Laplacian $L_{X}=D_{X}^{2}$, implying ${\rm str}(L^{k})=0$ for $k\geq 1$ so that ${\rm str}(\exp(-tL(X))={\rm str}(1_{X})=\chi(X)$.

In the quadratic cohomology case, where we look at functions on pairs of intersecting simplices, there is the cohomology of $G$ leading to $b(G)$ and five interaction cohomologies. They each lead to Betti vectors. We call them $b(K),b(U)$,$b(K,U),b(U,K)$, and $b(U,U)$. Besides pointing out that we have these new cohomologies, we give here a relation between them and the cohomology of $G$. We call it the quadratic fusion inequality. We could use the heavier notation $b(X,Y,Z)$ with $X,Y,Z\in\{K,U\}$ dealing with functions on pairs $(x,y)\in X\times Y$ with $x\cap y\in Z$, but we prefer to stick to the simpler notation: one reason is that both $b(U),b(K)$ deal with internal cohomology of $U$ and $K$ and do not involve simplices of the other set, while $b(K,U),b(U,K),b(U,U)$ involve both sets $U$ and $K$. So, while the quadratic Betti vectors $b(U),b(K)$ are intrinsic and only depend on one of the sets $U$ or $K$, the others are not. We prove:

The quadratic Betti vector $b(X)$ belongs to $\{(x,y),x\cap y\in X\}$, the vector $b(K,U)$ belongs to all $(x,y)\in K\times U$ with $x\cap y\in K$, and $b(U,K)$ belongs to all $(x,y)\in U\times K$ with $x\cap y\in K$, and $b(U,U)$ belongs to all $(x,y)\in U\times U$ with $x\cap y\in K$. Note that $x\cap y\in K$ if $x,y\in K$ so that $b(K,K)$ is accounted for in $b(K)$ already. With the heavier notation, from the eight cases $b(X,Y,Z)$ with $X,Y,Z\in\{K,U\}$ only $b(K)=b(K,K,K),b(U)=b(U,U,U),b(K,U)=b(K,U,K),b(U,K)=b(U,K,K),b(U,U)=b(U,U,K)$ can occur.

The quadratic characteristics are defined as $w(X)=\sum_{x,y\in X,x\cap y\in X}w(x)w(y)$ (Wu characteristic) and $w(X,Y)=\sum_{x\in X,y\in Y,x\cap y\in K}w(x)w(y)$ and $w(U,U)=\sum_{x\in U,y\in U,x\cap y\in U}w(x)w(y)$, we have $w(G)=w(U)+w(K)+w(U,K)+w(K,U)+w(U,U)$ which follows from the fact that $f$-vectors for quadratic cohomology satisfy $f(G)=f(K)+f(U)+f(K,U)+f(U,K)+f(U,U)$. Any of the cohomologies for $X\in\{G,U,K,(U,K),(K,U),(U,U)\}$ satisfy the Euler-Poincaré formula, following heat deformation with $L_{X}$ or $L_{X,Y}$ using the McKean-Singer symmetry.

Why is this interesting? If $G$ is finite abstract simplicial complex that is a finite $d$-manifold, and $f$ is an arbitrary function from $G$ to $P=\{0,\dots,k\}$, then the discrete Sard theorem [11] assures that the open set $U=\{x\in G,f(x)=P\}$ is either empty or a $(d-k)$-manifold in the sense that the graph with vertex set $U$ and edge set $\{(x,y)$ for which $x\subset y$ or $y\subset x\}$ is a discrete $(d-k)$-manifold. Since $U$ is open, the complement $K=G\setminus U$ is closed and so a sub-simplicial complex. All these cohomologies are topological invariants.

For example, if $G$ is a discrete $3$-sphere and $f:G\to\{0,1,2\}$ is arbitrary, then $U$ is either empty, a knot or a link, a finite union of closed disjoint (possibly interlinked) $1$-manifolds in the 3-sphere $G$. The simplicial cohomology of $U$ and the quadratic cohomology of $U$ are not interesting: they are just circles: $b(U)=(l,l)$, where $l$ is the number of connected components of $U$. The simplicial cohomology of $K$ however can be interesting and leads to knot or link invariants. It is well studied in the continuum as it is a knot invariant or a link invariant. Additional interaction cohomologies that take into account interaction between $U$ and the complement $K$ are completely unexplored. The inequality shows however that in general, more cohomology classes are created when splitting up $G$ into $U\cup K$. Unlike in the linear case, we have now the possibility of particles (harmonic forms) that are functions of $(x,y)$ with $x\in U,y\in K$ with $x\cap y\in K$ and also of function on $(x,y)$ with $x\in U,y\in U$ with $x\cap y\in K$.

The quadratic case we look at here would generalizes in a straightforward way to higher characteristics. One starts with $m=1$, the linear characteristic which is Euler characteristic. The second $m=2$ is quadratic characteristic or Wu characteristic going back to Wu in 1959. ¹¹1Historically, multi-linear valuations were considered in [14, 3, 6] and its cohomology in [8]. Quadratic cohomology is to Wu characteristic what simplicial cohomology is to Euler characteristic. In general, we would look for $m$-tuples of points $(x_{1},\dots,x_{m})$ that do simultaneously intersect. There are then much more $m$-point interaction cohomologies and $b(G)$ is again bounded above by all possible cases of $b(X_{1},\dots,X_{m})$ with $X_{i}\in\{U,K\}$ and the intersection in $K$. The cases $b(U,U,\dots,U)$ has cases $(x_{1},\dots x_{m})\in U^{m}$ with $\bigcap x_{i}\in U$ which is part of $b_{U}$ and a new part where $\bigcap x_{i}\in K$. As in the case $m=2$, the cohomology of $b(K,K,\dots,K)$ is part of $b(K)$. We have to consider the Betti vector $b(U)$ for functions on $\{(x_{1},\dots,x_{m})\in U^{m},\bigcap_{i}x_{i}\in U\}$ and $b(U,\dots,U)$ referring to functions on $\{(x_{1},\dots,x_{m})\in U^{m},\bigcap_{i}x_{i}\in K\}$, where $b(K,\dots,K)=b(K)$ in the notation used before. If all $x_{i}$ are in $U$, then the intersection can be either in $U$ or $K$, but if all $x_{i}$ are in $K$, then the intersection must be in $K$, not warranting to distinguish $b(K,\dots,K)$ and $b(K)$. Despite the obvious duality $U\leftrightarrow K$, there is an asymmetry in that $x\in U,y\in U$ allows $x\cap y$ to be in $U$ or $K$ while $x\in K,y\in K$ implies $x\cap y\in K$: technically, a closed set $K$ is a $\pi$-system while an open set $U$ is not unless it is $\emptyset$ or $G$.

There would again be heavier notation $b(X_{1},\dots,X_{m},X_{0})$ dealing with $(x_{1},\dots,x_{m})\in X_{1}\times\cdots\times X_{m}$ with $\bigcap_{k=1}^{m}x_{k}\in X_{0}$ and have $b(G)\leq\sum_{X_{i}\in\{U,K\}}b(X_{1},\dots,X_{m},X_{0})$ taking into account that some of the cases like $(K,\dots,K,U)$ are empty because $X_{0}=U$ only is interesting if all $X_{1}=\cdots=X_{m}=K$. Again, like in the case $m=2$, we have two cases $b(U),b(K)$ which are intrinsic while the other cases involve simplices from both $U$ and $K$.

We study here higher order chain complexes in finite geometries. Each of the situations is defined by a triple $(X,D,R)$, where $X$ is a finite set of $n$ elements, $D=d+d^{*}$ is a finite $n\times n$ matrix such that $d^{2}=0$ and where $R$ is the dimension function compatible with $D$ in the sense that the blocks of $L=D^{2}$ have constant dimension. We can call this structure an abstract delta set because every delta set defines such a structure, but where instead of face maps, we just go directly to the exterior derivative $d$. The advantage of looking at the Dirac setting is that $D$ can be much more general than coming from face maps. It could be a deformed Dirac matrix for example obtained by isospectral deformation $D^{\prime}=[D^{+}-D^{-},D]$ [5, 4], which keeps the spectrum of $D$ invariant but produces $D=d+d^{*}+B$ leading to new exterior derivatives $d(t)$, a deformation which is invisible to the Hodge Laplacian as $D^{2}(t)=L$ is not affected. Since $d(t)$ gets smaller, this produces an expansion of space. In general, also in the continuum, there is an inflationary start of expansion.

Lets illustrate the quadratic fusion inequality in the case $K_{2}$:

The linear simplicial cohomology is given by $(\left[\begin{array}[]{c}\{1\}\\ \{2\}\\ \{1,2\}\end{array}\right],\left[\begin{array}[]{ccc}0&0&-1\\ 0&0&1\\ -1&1&0\\ \end{array}\right],R=\left[\begin{array}[]{c}0\\ 0\\ 1\end{array}\right])$. Take the open-closed pair $U=\{\{1,2\}\}$ and $K=\{\{1\},\{2\}\}$ leading to the abstract delta set structures $(\left[\begin{array}[]{c}\{1\}\\ \{2\}\end{array}\right],\left[\begin{array}[]{cc}0&0\\ 0&0\\ \end{array}\right],\left[\begin{array}[]{c}0\\ 0\end{array}\right])$ $([\{1,2\}],\left[\begin{array}[]{c}0\\ \end{array}\right],[1])$. The Betti vectors are $b(G)=(1,0),b(U)=(0,1),b(K)=(2,0)$ and the f-vectors are $f(G)=(2,1),f(U)=(0,1),f(K)=(2,0)$. The fusion inequality $b(G)<b(K)+b(U)$ is here strict. Merging $U$ and $K$ fuses a harmonic $0$ form in $K$ with the $1$-form in $U$. Betti vectors have been considered since Betti and Poincaré. Finite topological spaces were first looked at by Alexandroff [1]. For cohomology of open sets in finite frame works, see [10].

If we look at quadratic cohomology for $G$, where we have the abstract delta set $(X,D,R)=$

The Hodge Laplacian $L=D^{2}=L_{0}\oplus L_{1}\oplus L_{2}$ has the Hodge blocks:

with Betti vector $b(G)=(0,1,0)$ and f-vector $f(G)=(2,4,1)$ and Wu characteristic $w(G)=f_{0}-f_{1}+f_{2}=2-4+1=b_{0}-b_{1}+b_{2}=0-1+0=-1$. The eigenvalues of $L_{1}$ are $(0,2,2,4)$, the null-space is spanned by $[1,1,1,1]$. By accident $L_{1}$ happens to be a Kirchhoff matrix of $C_{4}$. If $K_{2}$ is seen as a 1-manifold with boundary $\delta G$ ²²2We usually assume that manifolds with boundary have an interior. The Barycentric refinement of a complete graph $K_{d+1}$ would be a $d$-manifold with boundary. we have $w(G)=\chi(G)-\chi(\delta(G))=1-2=-1$, illustrating that in general, for manifolds $M$ with boundary $\delta M$, the Wu characteristic is $\chi(M)-\chi(\delta M)$.

Now to $U=\{\{1,2\}\}$, where we have the abstract delta set structure $(X,D,R)=$

with $b(U)=(0,0,1)$ and $f(U)=(0,0,1)$ and $w(U)=1$.
For $K=\{\{1\},\{2\}\}$ we have the abstract delta set structure $(X,D,R)=$

with $b(K)=(2,0,0)$ and $f(U)=(2,0,0)$ and $w(K)=2$. Obviously the intrinsic cohomologies of $U$ and $K$ are not yet giving a complete picture. The simplices in $U$ and $K$ can interact as we see next.

Now, we turn to the interactions of $K$ with $U$

with $b(K,U)=b(U,K)=(0,2,0)$ and $f(U)=(0,2,0)$ and $w(U,K)=-2$. There is no pair $(x,y)\in U^{2}$ such that $x\cap y\in K$ so that $b(U,U)=(0,0,0)$.

To summarize, we have

The quadratic fusion inequality $b(U)+b(K)+b(U,K)+b(K,U)+b(U,U)=(2,3,1)>b(G)=(0,1,0)$ is here strict. The fusion has two $0$-form-$1$-form mergers and one $1$-form-$2$-form merger. The difference in the fusion inequality is is $(1,1,0)+(1,1,0)+(0,1,1)=(2,3,1)$.

We see already in this small example, how the closed “laboratory” $K$ and the “observer space” $U$ are no more strictly separated, even so they partition the “world” $G$. The “tunneling” between $K$ and $U$ is described using algebraic topology, expressed by cohomology groups. Unlike for simplicial cohomology which features homotopy invariance, there is only topological invariance. Already the Wu characteristic of contractible balls depends on the dimension. [For a $d$-ball, the Wu characteristic $w(M)$ is $(-1)^{d}$ where $d$ is the dimension and illustrates that $w(M)=\chi(M)-\chi(\delta M)$ in general for discrete manifolds $M$ with boundary $\delta M$ and that for a $d$-ball, the boundary is a $d-1$ sphere with Euler characteristic $\chi(\delta M)=1-(-1)^{d}$.] If we take a $d$-ball in a $d$-dimensional simplicial complex and replace the interior to get an other d-ball without changing the boundary, then the cohomology does not change because we can for any positive $k$ add add gauge fields (k-forms that are coboundaries) $dg$ to render a cocycle zero in the interior (without changing the equivalence class) and use the heat flow to get back a harmonic form after doing the surgery in the interior. This implements the chain homotopy when doing a local homeomorphic deformation: move the field away from the “surgergy place”, do the surgery, then use the heat flow to “heal the wound” and get back harmonic forms.

We have just given the argument for the following result:

For $G=K_{3},K=\{\{1\}\}$, we can look at the complex $G(U,K)=\left[\begin{array}[]{cc}\{1\}&\{1,3\}\\ \{1\}&\{1,2\}\\ \{1\}&\{1,2,3\}\\ \end{array}\right]$ and $D(K,U)=\left[\begin{array}[]{ccc}0&0&-1\\ 0&0&1\\ -1&1&0\\ \end{array}\right]$ with kernel spanned by $[1,1,0]$. For its Barycentric refinement and still $K=\{1\}$ (on the boundary), and where we look at functions on $X=\left[\begin{array}[]{cc}\{1\}&\{1,7\}\\ \{1\}&\{1,5\}\\ \{1\}&\{1,4\}\\ \{1\}&\{1,5,7\}\\ \{1\}&\{1,4,7\}\\ \end{array}\right]$, we have $D(K,U)=\left[\begin{array}[]{ccccc}0&0&0&-1&-1\\ 0&0&0&1&0\\ 0&0&0&0&1\\ -1&1&0&0&0\\ -1&0&1&0&0\\ \end{array}\right]$ with kernel spanned by $[1,1,1,0,0]$.

Simplicial cohomology for a finite abstract simplicial complex $G$ is part of the spectral theory of the Hodge Laplacian $L=D^{2}$ with Dirac matrix $D=d+d^{*}$, where $d$ is the exterior derivative. Note that all these matrices $d,D,L$ are $n\times n$ matrices if $G$ has $n$ elements. The matrix $L$ is a block diagonal matrix $L=\oplus_{k=0}^{d}L_{k}$. The kernels of the blocks $L_{k}$ of $L$ are the $k$-harmonic forms or $k$-cohomology vector spaces. In this finite setting, this is linear algebra [2]. The dimensions $b_{k}$ are the Betti numbers, the components of the Betti vector of $G$.

If $K$ is a subcomplex of $G$ and $U$ is the open complement, then the separated system $(K,U)$ has a Laplacian $L_{K,U}=L_{K}\oplus L_{U}$ for which the energies $\lambda_{j}(L_{K,U})$ are less or equal than $2\lambda_{j}(L_{G})$ [13] implying that the separated system can not have more harmonic forms than $G$. It can have more: if $G$ is a closed 2-ball for example and $K$ is the boundary 1-sphere then $b_{G}=(1,0,0),b_{K}=(1,1,0)$ and $b_{U}=(0,0,1)$. The closed part $K$ carries a trapped harmonic 1-form. It is fused with the 2-form present on $U$, if $K,U$ get united to $G$.

A complex $G$ defines a delta set $G=\bigcup_{k=0}^{d}G_{k}$. The $f$-vector $f(G)=(f_{0}(G),\dots,f_{d}(G))$ has components $f_{k}(G)=|G_{k}|$, the number of elements in $G_{k}$. The super trace of an $n\times n$ matrix $L$ ³³3We write the entries as $L(x,y)$ is defined as ${\rm str}(L)=\sum_{k=0}^{d}(-1)^{k}\sum_{x\in G_{k}}L(x,x)$. Compare with the usual trace ${\rm tr}(L)=\sum_{k=0}^{d}\sum_{x\in G_{k}}L(x,x)=\sum_{x\in G}L(x,x)$. The Euler characteristic is $\chi(G)=\sum_{x\in G}w(x)$. The Euler-Poincaré formula $\chi(G)=\sum_{k}(-1)^{k}f_{k}=\sum_{k}(-1)^{k}b_{k}$ follows directly from the McKean-Singer identity, stating that ${\rm str}(\exp(-itL))=\chi(G)$ for all $t$ which in turn follows from the fact that the Dirac matrix $D$ gives an isomorphism between even and odd non-harmonic forms. For $t=0$, the super trace of the heat kernel is the combinatorial Euler characteristic, while for $t=\infty$, it is the cohomological Euler characteristic.

Quadratic cohomology does not build on single simplices $x\in G$ like simplicial cohomology but on pairs of intersecting simplices $(x,y)\in G\times G$. Define $w(x)=(-1)^{{\rm dim}(x)}$. The quadratic analog of (linear) Euler characteristic $\chi(A)=\sum_{x\in A}w(x)$ is the “Ising type” energy or Wu characteristic $w(A)=\sum_{x,y,x\cap y\in A}w(x)w(y)$. It is an example of a multi-linear valuation. We also just call it quadratic characteristic, an example of higher characteristic. [9].

The name “quadratic” is chosen because it is multi-linear and for $m=2$ a quadratic valuation. Similarly as a quadratic form is a multi-linear map, linear in each argument, the quadratic characteristic $w(A,B)=\sum_{x\in A,y\in B,x\cap y\neq\emptyset}w(A)w(B)$ (or variants, where we ask the intersection to be in $A$ or $B$) satisfies the valuation formula in each of the coordinates, like $w(X,U\cup V)=w(X,U)+w(X,V)-W(X,U\cap V)$.

Given an open-closed pair $(U,K)$, one can define quadratic cohomology on $k$-forms. Forms are functions on $\Lambda(X,Y)=\{(x,y)|x\in X,y\in Y,x\cap y\in X\}$ and $\Lambda(X)=\{(x,y)|x\in X,y\in X,x\cap y\in X\}$ and $\Lambda(X,Y)=\{(x,y)|x\in X,y\in Y,x\cap y\neq\emptyset,x\cap y\in K\}$. The $k$-forms are the forms on functions with ${\rm dim}(x)+{\rm dim}(y)=k$.

In the case of an open-closed pair, we have five different cohomologies $U,K$, $(U,K)$, $(K,U)$, $(U,U)$. There is no case $(K,K)$ because the intersection of $x\in K,y\in K$ is in $K$. The case $(K,K)$ is part of $K$. The case $(U,U)$ looks at pairs such that the intersection in in $K$ while $U$ looks at pairs such that the intersection is in $K$. We can have a disjoint union

The exterior derivative is inherited from the exterior derivative on products. It is $df(x,y)=d_{x}f(x,y)+w(x)d_{y}(f,y)$, where $d_{x},d_{y}$ are the usual simplicial exterior derivatives but with respect to the first or second coordinate. If we would look at this derivative on $X\times Y$, the Hodge Laplacians are the tensor products of the Laplacians on $X$ and $Y$. Even if the set-theoretical Cartesian product $X\times Y$ is not a simplicial complex any more, we still have a cohomology. But now, we restrict this exterior derivative to pairs $(x,y)$ that intersect. We are not aware of such a construction in the continuum.

The proof of the quadratic fusion inequality Theorem (2) is analog to the linear case. The key is that in each case, the matrix $L$ is the square $L=D^{2}$ of a matrix $D$ which has the property that a principal sub-matrix of $D$ has intertwined spectrum so that the left padded spectral functions of $L$ are monotone. This looks like a technical detail but it is important and at the heart of the entire story: the matrix $L$ does not have the property that taking away highest or lowest dimensional simplices produces principal sub-matrices which themselves come from a geometry. But the Dirac matrix $D$ does have the property. And since $D$ has symmetric spectrum with respect to the origin and $D^{2}=L$, we have also monotonocity for $L$.

Let us formulate the Cauchy interlace theorem a bit differently, than usual. The point is that if a principal submatrix $B$ of a self-adjoint matrix $A$ has the eigenvalues padded left when compared to the eigenvalues of $A$ then there is a direct comparison between all eigenvalues. This is very general and allows to talk about monotonicity rather than interlacing.

We can now look at the map $K\to\lambda(K)$ giving for each sub-complex $K$ the spectral function ordered in an ascending way and padded left. The partial order on sub-simplicial complexes and the partial order on spectral functions are compatible:

The same holds by iterating the process and take principal $(n-k)\times(n-k)$ sub-matrices. Now, if we look at a Dirac matrix of a closed set $K$ and take a maximal simplex $x$ away, then we get monotonicity. The same happens if we take a minimal simplex $x$ away from an open set. Note that if we look at pairs $(x,y)$ belonging to some pair like $(K,U)$ and we take a maximal element $x$ away, then several pairs $(x,y_{i})$ are removed from the complex on $(K,U)$.

In the quadratic case, taking a way a largest dimensional simplex (facet) $x$ will affect in general various pairs of simplices $(x,y)$ or $(y,x)$. The effect is that the quadratic Dirac matrix of $G\setminus x$ is still a principal sub-matrix. We still have spectral monotonicity.

To conclude the proof of Theorem (2), write down the decoupled Laplacian $L(U)\oplus L(K)\oplus L(K,U)\oplus L(U,K)\oplus L(U,U)$ which is block diagonal and is a $n\times n$ matrix, the same size than $L(G)$. Lets call its eigenvalues $\mu_{k}$. From the spectral inequalities for each block, we know

where $\lambda_{k}$ are the eigenvalues of the quadratic Hodge Laplacian of $G$. Therefore, there are at least as many 0 eigenvalues for the decoupled system than for $G$, proving the inequality.

Here is an example with the Kite complex $G=K_{1,2,1}$, where we have a complex with 2 triangles. We will see what happens if we take one of the triangles away. We look at the case $(U,U)$. The Dirac matrix $D(U,U)$ is a $14\times 14$ matrix.