We construct homology theories with coefficients in L-spectra on the category of ball complexes and we define products in this setting. We also obtain signatures of geometric situations in these homology groups and prove product formulae which we hope will clarify products used in the theory of the total surgery obstruction.

We construct homology theories with coefficients in L-spectra on the category of ball complexes and we define products in this setting. We also obtain signatures of geometric situations in these homology groups and prove product formulae which we hope will clarify products used in the theory of the total surgery obstruction.

A map f : X → Y to a simplicial complex Y is called a Y -triangular homotopy equivalence if it has a homotopy inverse g and homotopies h1 : and homotopies h1|σ and h2|σ. In this paper we prove that for all pairs X, Y of finite-dimensional locally finite simplicial complexes there is an ǫ(X, Y ) > 0 such that any ǫ-controlled homotopy equivalence f : X → Y for ǫ < ǫ(X, Y ) is homotopic to a Y -triangular homotopy equivalence. Conversely, we conjecture that it is possible to 'subdivide' a Y -triangular homotopy equivalence by finding a homotopic (Sd Y )-triangular homotopy equivalence, consequently a Y -triangular homotopy equivalence would be homotopic to an ǫ-controlled homotopy equivalence for all ǫ > 0.