Fusion inequality for quadratic cohomology

Abstract

Classical simplicial cohomology on a simplicial complex G deals with functions on simplices x in G. Quadratic cohomology deals with functions on pairs of simplices (x,y) in G x G that intersect. If K,U is a closed-open pair in G, we prove here a quadratic version of the linear fusion inequality. Additional to the quadratic cohomology of G there are five additional interaction cohomology groups. Their Betti numbers are computed from functions on pairs (x,y) of simplices that intersect. Define the Betti vector b(X) computed from pairs (x,y) in X x X with x intersected y in X a and b(X,Y) with pairs in X xY with x intersected y in K. We prove the fusion inequality b(G) <= b(K)+b(U)+b(K,U)+b(U,K)+b(U,U) for cohomology groups linking all five possible interaction cases. Counting shows f(G) = f(K)+f(U) + f(K,U)+f(U,K)+f(U,U) for the f-vectors. Super counting gives Euler-Poincare sum_k (-1)^k f_k(X)=\sum_k (-1)^k b_k(X) and sum_k (-1)^k f_k(X,Y)=sum_k (-1)^k b_k(X,Y) for X,Y in {U,K}. As in the linear case, also the proof of the quadratic fusion inequality follows from the fact that the spectra of all the involved Laplacians L(X),L(X,Y) are bounded above by the spectrum of the quadratic Hodge Laplacian L(G) of G.

Figures (3)

• Figure 1: We see the sorted eigenvalues of a random real self-adjoint $300 \times 300$ matrix $A_0$, then the eigenvalues of a $200 \times 200$ principal submatrix $A_1$ and then the eigenvalues of a $100 \times 100$ principal submatrix $A_2$ of $A_1$. The eigenvalues are padded left. The figure illustrates Lemma (\ref{['leftpaddedmonotonicity']}). The code which gave the output is listed below.
• Figure 2: The Dirac matrix $D$ and the Hodge Laplacian $L=D^2$ in the linear case for the kite graph $G$. The splittings are given by the f-vector $f(G)=(4,5,2)$. There are 4 points, 5 edges and 2 triangles in $G$.
• Figure 3: The Dirac matrix $D$ and the Hodge Laplacian $L=D^2$ in the quadratic case for the kite graph. The splittings are given by the f-vector $f(G)=(4,20,33,20,4)$. The space of $1$-forms (intersecting points) is $4$-dimensional, the space of $2$-forms (intersection of a point with an edges) is $20$-dimensional, the space of $3$-forms (intersection of two edges or a triangle-point has dimension $33$), the space of $4$ forms (intersection of an edge and triangle) is $20$-dimensional, the space of $5$ forms (intersection of two triangles) is $4$-dimensional.

Theorems & Definitions (9)

• Theorem 1: Spectral inequality
• Theorem 2: Quadratic fusion inequality
• Theorem 3
• Theorem 4
• Lemma 1: Left Padded Monotonicity
• proof
• Corollary 1
• Theorem 5: Spectral monotonicity
• proof