  Let $\Mg$ denote the moduli space of compact Riemann surfaces of genus $g$.
Mumford had proved that, for each fixed genus $g$, there are isomorphisms
asserting that certain higher $DET$ bundles over $\Mg$ are certain fixed
(genus-independent) tensor powers of the Hodge line bundle on $\Mg$. Here we
obtain a coherent, genus-independent description of the Mumford isomorphisms
over certain infinite-dimensional ``universal'' parameter spaces of compact
Riemann surfaces (with or without marked points). We work with an inductive
limit of Teichm\"uller spaces comprising complex structures on a certain
``solenoidal Riemann surface'', $H_{\infty,abel}$. The main result constructs
these universal $DET$ and Hodge line bundles on this direct limit, equips them
with their Quillen metrics, and proves that they are related by the appropriate
Mumford isomorphisms. Our work can be viewed as a contribution towards a
non-perturbative formulation of the bosonic string theory, since it provides an
interpretation of the Polyakov measure in a genus-independent fashion.
