Theorem. Harmonic cochains realize the Betti numbers [fdip-0090]
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  • November 2, 2026

Let the square torus, its coboundary \(d\), and its evaluation pairing be those of the Betti count. There is no cell below a vertex or above a face, so the coboundary of a face function is zero and there is no adjoint into a vertex from a lower cell. Write \(d^\dagger \) for the adjoint of each coboundary in the evaluation pairing: \(\langle d\gamma ,\beta \rangle =\langle \gamma ,d^\dagger \beta \rangle \). On edge fields this is the counting adjoint. The pairing is symmetric, so \(d\) and \(d^\dagger \) are adjoints of one another, and \(d+d^\dagger \) is self-adjoint. The adjoint of the zero map is zero, and the adjoint reverses composition, so \(d\circ d=0\) gives \(d^\dagger \circ d^\dagger =0\).

On a cochain \(\gamma \) of one degree, define \(\Delta \gamma =d(d^\dagger \gamma )+d^\dagger (d\gamma )\), with either summand omitted when its map has no cells to land on. Call \(\Delta \) the Hodge Laplacian, and call \(\gamma \) harmonic when \(\Delta \gamma =0\). Then \(\Delta =(d+d^\dagger )\circ (d+d^\dagger )\) on each degree. On vertex functions, and with every edge weight equal to \(1\), \(\Delta \) is the unit-weight case of the weighted graph Laplacian. The harmonic cochains in each degree realize the cohomology of that degree, so their dimensions are the Betti numbers \((1,2,1)\). The kernel of \(d+d^\dagger \) therefore has dimension \(4\). A vertex function is the adjoint of an edge field if and only if the sum of its values is zero.

Proof. 

  • November 2, 2026

On a cochain of one degree the two summands of \((d+d^\dagger )\circ (d+d^\dagger )\) are \(d(d^\dagger \gamma )\) and \(d^\dagger (d\gamma )\), because \(d\circ d\) and \(d^\dagger \circ d^\dagger \) vanish. Thus the square is \(\Delta \). Self-adjointness gives \(\langle \Delta \Phi ,\Phi \rangle =\langle (d+d^\dagger )\Phi ,(d+d^\dagger )\Phi \rangle \) for a sum \(\Phi \) of cochains of all three degrees. Expanding the degree-wise formula gives \[\langle \Delta \gamma ,\gamma \rangle =\langle d\gamma ,d\gamma \rangle +\langle d^\dagger \gamma ,d^\dagger \gamma \rangle .\] Both summands are nonnegative, so \(\Delta \gamma =0\) if and only if \(d\gamma =0\) and \(d^\dagger \gamma =0\). The same identity shows that \((d+d^\dagger )\Phi =0\) if and only if every degree of \(\Phi \) is harmonic.

On vertex functions the incoming adjoint is zero, so \(\Delta f=d^\dagger (df)\). The counting adjoint on edge fields is this \(d^\dagger \), and for the constant weight \(1\) the weighted graph Laplacian is \(d_R^\dagger d_R\). The two operators agree.

Fix a degree, and write \(d_-\) for the coboundary into that degree. Its image is contained in the kernel of the outgoing coboundary, because \(d\circ d=0\). The rank of \(d_-^\dagger \) equals the rank of \(d_-\), so the kernel of the adjoint has dimension equal to the dimension of the cochain space minus that rank. Added to the dimension of the image, the total is the whole space. If \(d_-\alpha \) lies in the kernel of \(d_-^\dagger \), then \(\langle d_-\alpha ,d_-\alpha \rangle =\langle \alpha ,d_-^\dagger d_-\alpha \rangle =0\), so \(d_-\alpha =0\). The image therefore meets that kernel only at zero, and the cochain space is their direct sum. A closed cochain splits uniquely as \(d_-\alpha +\rho \) with \(d_-^\dagger \rho =0\). The outgoing coboundary kills \(d_-\alpha \) and kills the sum, so it kills \(\rho \). Thus \(\rho \) is harmonic. The splitting of the closed cochains is direct, and the dimension of the harmonic space is the Betti number of that degree.

On vertices the kernel of \(d\) is the constants. The Betti count gives that kernel dimension \(1\), so the rank of \(d\) is one less than the number of vertices. The image of \(d^\dagger \) from edge fields is orthogonal to that kernel, because \(\langle d^\dagger \alpha ,f\rangle =\langle \alpha ,df\rangle \). The rank of this adjoint equals the rank of \(d\), so the image is the orthogonal complement of the constants. The pairing of a vertex function against the constant \(1\) is the sum of its values, and therefore the complement is the set of vertex functions whose values sum to zero.

The constant vertex functions are harmonic. The two winding edge fields of the Betti count are closed, and each vertex is the head of one edge and the source of one edge in the direction of the winding, with equal values, so the counting-adjoint formula gives zero. The sum of the face values of every coboundary is zero, so the constant face function is orthogonal to every coboundary and its adjoint vanishes. The 4 cochains lie in the three degrees, and the two edge fields are not scalar multiples of one another, so they are linearly independent. The kernel has dimension \(4\), and therefore they form a basis.