Template:Kreck1999

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

1 Comments

We consider the question of the surgery obstruction $\Theta_W$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}\Theta_W$ for a bordism $(W; M_0, M_1, \bar \nu)$$(W; M_0, M_1, \bar \nu)$ as in Theorem 3 but when the maps $\bar \nu_i \colon M_i \to B$$\bar \nu_i \colon M_i \to B$ are normal $(q-1)$$(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$$\langle w_{q+1}(B), \pi_{q+1}(B) \rangle = 0$, then

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

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

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

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

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

In particular, there are isomorphisms

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

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/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_AKd7C1$$B$ of a given manifold $M$$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$$F_n$' if it admits an $n/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_6XctWL$$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.