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Invariance of Torsion and the Borsuk Conjecture

The following results of Whitehead and Wall are well-known applications of the algebraic K-theoretic functors K 0 and K 1 to basic homotopy questions in topology. THEOREM 1 [20]. If f : X → Y is a homotopy equivalence between compact CW complexes, then there is a torsion τ(ƒ) in the algebraically-defined Whitehead group Wh π1(Y) which vanishes if and only if f is a simple homotopy equivalence. THEOREM 2 [18]. If X is an arbitrary space which is finitely dominated (i.e., homotopically dominated by a compact polyhedron), then there is an obstruction σ(X) in the algebraically-defined reduced projective class group which vanishes if and only if X is homotopy equivalent to some compact polyhedron. If we direct sum over components, then the above statements make good sense even if the spaces involved are not connected.