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M. Kreck, Surgery and duality, Ann. of Math. (2) 149 (1999), no.3, 707–754. MR1709301 (2001a:57051) Zbl 0935.57039

[edit] 1 Comments

We consider the question of the surgery obstruction \Theta_W for a bordism (W; M_0, M_1, \bar \nu) as in Theorem 3 but when the maps \bar \nu_i \colon M_i \to B are normal (q-1)-smoothings. In this case, the proof of Theorem 3 shows that if \langle w_{q+1}(B), \pi_{q+1}(B) \rangle = 0, then

\displaystyle  \Theta_W \in L_{2q}(\pi_1(B), w_1(B)) \subset l_{2q+2}(\pi_1(B), w_1(B)),

and if \langle w_{q+1}(B), \pi_{q+1}(B) \rangle \neq 0, then

\displaystyle  \Theta_W \in \widetilde L_{2q}(\pi_1(B), w_1(B)) \subset \widetilde l_{2q+2}(\pi_1(B), w_1(B)).

Here L_{2q+2}(\pi_1(B), w_1(B)) and \widetilde L_{2q+2}(\pi_1(B), w_1(B)) may be defined as the groups of units in l_{2q+2}(\pi_1(B), w_1(B)) and \widetilde l_{2q+2}(\pi_1(B), w_1(B)) respectively. Alternatively, they may be defined as certain Witt groups of quadratic forms over the twisted group ring (\Zz[\pi_1(B)], w_1(B)). In the case of the tilde groups, when q = 2, 6, one simply considers quadratic forms with values in

\displaystyle  Q_{-1}(\Zz[\pi_1(B)], w_1(B))/Q_{-1}(e).

In particular, there are isomorphisms

\displaystyle  \widetilde L_{2q+2}(e) \cong L^{2q+2}(e) \cong 0,

where L^{2q+2}(e) is the symmetric L-group.

[edit] 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 \pi_3(S^1 \vee S^2), 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 W into the normal 1-type B of a given manifold M can be made into a 3-equivalence by proposition 4, but this is not generally true: One way to formulate the assumption of Proposition 4 for achieving this is that B be of type F_3 (generally, one calls a space X 'of type F_n' if it admits an n-equivalence from a finite cell complex). However, while B is of course of type F_2, it is not obviously of type F_3 and indeed it need not be: From B one still has a canonical 2-equivalence to B\pi_1(M) which is a 3-equivalence if the universal cover of M admits a spin-structure. If B is of type F_3 this exhibits B\pi_1(M) as being of type F_3 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 F_2 and it is generally true, that there are groups whose classifying spaces are of type F_n but not F_{n+1}. To correct the argument one can use proposition 4 to produce a 2-equivalence from out of the map from W to B by surgeries in the interior of W and then to surgeries by hand to make the inclusion M \rightarrow W surjective on \pi_2. That this is indeed possible is proven in Theorem 2.2 of Rosenberg's paper 'C^*-algebras, positive scalar curvature, and the Novikov conjecture II' [Rosenberg1986] for the case of spin-manifolds and the proof immediately generalises.

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