1 Comments
We consider the question of the surgery obstruction for a bordism as in Theorem 3 but when the maps are normal -smoothings. In this case, the proof of Theorem 3 shows that if , then
and if , then
Here and may be defined as the groups of units in and respectively. Alternatively, they may be defined as certain Witt groups of quadratic forms over the twisted group ring . In the case of the tilde groups, when , one simply considers quadratic forms with values in
In particular, there are isomorphisms
where is the symmetric -group.
 2 Corrections
While their statements are correct, the proofs of both Proposition 4 and Theorem 1 of Section 3 are incorrect:
The supplied proof of Proposition 4 uses (on p. 718, right before Lemma 3) the fact that a finite cell complex has its higher homotopy groups finitely generated over its fundamental group-ring. This, however, is not the case in general, probably the easiest counterexample being , due to the occurrence of Whitehead-products. It is in fact true that the lowest homotopy group of a map between finite cell complexes is finitely generated over the fundamental group ring of the source, once it induces an isomorphism on the fundamental group. This suffices to successively make a given normal map highly connected in the setup of proposition 4. Details can easily be adapted from Lück's Trieste notes.
The mistake in the proof of Theorem 1 is a simply misquote of Proposition 4: It is claimed that a normal map from a given bordism into the normal -type of a given manifold can be made into a -equivalence by proposition 4, but this is not generally true: One way to formulate the assumption of Proposition 4 for achieving this is that be of type (generally, one calls a space 'of type ' if it admits an -equivalence from a finite cell complex). However, while is of course of type , it is not obviously of type and indeed it need not be: From one still has a canonical -equivalence to which is a -equivalence if the universal cover of admits a spin-structure. If is of type this exhibits as being of type as well. However, it is well-known that an arbitrary finitely presented group can arise as the fundamental group of a closed, (spin!?)-manifold. Finitely presented groups are precisely those, whose classifying spaces are of type and it is generally true, that there are groups whose classifying spaces are of type but not . To correct the argument one can use proposition 4 to produce a -equivalence from out of the map from to by surgeries in the interior of and then to surgeries by hand to make the inclusion surjective on . That this is indeed possible is proven in Theorem 2.2 of Rosenberg's paper '-algebras, positive scalar curvature, and the Novikov conjecture II' [Rosenberg1986] for the case of spin-manifolds and the proof immediately generalises.