Surjectivity of the algebraic assembly map onto K_*^{alg}(B M)

Ascertain whether every class in K_*^{alg}(B M) is in the image of the algebraic assembly map A^{alg}: {summable even/odd Fredholm modules} → K_*^{alg}(B M), i.e., determine whether A^{alg} is surjective for the algebra B M of locally trace-class, finite propagation operators on a proper metric space M.

Background

To relate the algebraic and analytic frameworks, the authors introduce an algebraic assembly map A^{alg} that assigns to a summable Fredholm module a class in K_*^{alg}(B M). Commutativity of key diagrams—and thus equality of pairings—can be established on the image of A^{alg}.

However, the approach hinges on whether all classes in K_*^{alg}(B M) arise from summable Fredholm modules. If A^{alg} were surjective, the main equality would extend more broadly; the authors note this surjectivity is not currently established.