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We revisit a construction principle of Fredholm operators using Hilbertcomplexes of densely defined, closed linear operators and apply this toparticular choices of differential operators. The resulting index is thencomputed with the help of explicitly describing the dimension of the cohomologygroups of generalised (`harmonic') Dirichlet and Neumann tensor fields. Themain results of this contribution are the computation of the indices ofDirac-type operators associated to the elasticity complex and the newly foundbiharmonic complex, relevant for the biharmonic equation, elasticity, and forthe theory of general relativity. The differential operators are of mixed orderand cannot be seen as leading order type with relatively compact perturbation.As a by-product we present a comprehensive description of the underlyinggeneralised Dirichlet-Neumann vector and tensor fields defining the respectivecohomology groups, including an explicit construction of bases in terms oftopological invariants, which are of both analytical and numerical interest.Though being defined by certain projection mechanisms, we shall present a wayof computing these basis functions by solving certain PDEs given in variationalform. For all of this we rephrase core arguments in the work of Rainer Picard[1982] applied to the de Rham complex and use them as a blueprint for the moreinvolved cases presented here. In passing, we also provide newvector-analytical estimates of generalised Poincare-Friedrichs type useful forelasticity or the theory of general relativity.

The Index of Some Mixed Order Dirac-Type Operators and Generalised Dirichlet-Neumann Tensor Fields