# Template:Kreck1999

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


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

and if $\langle w_{q+1}(B), \pi_{q+1}(B) \rangle \neq 0$$\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))$$L_{2q+2}(\pi_1(B), w_1(B))$ and $\widetilde L_{2q+2}(\pi_1(B), w_1(B))$$\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))$$l_{2q+2}(\pi_1(B), w_1(B))$ and $\widetilde l_{2q+2}(\pi_1(B), w_1(B))$$\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))$$(\Zz[\pi_1(B)], w_1(B))$. In the case of the tilde groups, when $q = 2, 6$$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)$$L^{2q+2}(e)$ is the symmetric $L$$L$-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 $\pi_3(S^1 \vee S^2)$$\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$$W$ into the normal $1$$1$-type $B$$B$ of a given manifold $M$$M$ can be made into a $3$$3$-equivalence by proposition 4, but this is not generally true: One way to formulate the assumption of Proposition 4 for achieving this is that $B$$B$ be of type $F_3$$F_3$ (generally, one calls a space $X$$X$ 'of type $F_n$$F_n$' if it admits an $n$$n$-equivalence from a finite cell complex). However, while $B$$B$ is of course of type $F_2$$F_2$, it is not obviously of type $F_3$$F_3$ and indeed it need not be: From $B$$B$ one still has a canonical $2$$2$-equivalence to $B\pi_1(M)$$B\pi_1(M)$ which is a $3$$3$-equivalence if the universal cover of $M$$M$ admits a spin-structure. If $B$$B$ is of type $F_3$$F_3$ this exhibits $B\pi_1(M)$$B\pi_1(M)$ as being of type $F_3$$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$$F_2$ and it is generally true, that there are groups whose classifying spaces are of type $F_n$$F_n$ but not $F_{n+1}$$F_{n+1}$. To correct the argument one can use proposition 4 to produce a $2$$2$-equivalence from out of the map from $W$$W$ to $B$$B$ by surgeries in the interior of $W$$W$ and then to surgeries by hand to make the inclusion $M \rightarrow W$$M \rightarrow W$ surjective on $\pi_2$$\pi_2$. That this is indeed possible is proven in Theorem 2.2 of Rosenberg's paper '$C^*$$C^*$-algebras, positive scalar curvature, and the Novikov conjecture II' [Rosenberg1986] for the case of spin-manifolds and the proof immediately generalises.