The stability theorem for smooth pseudoisotopies.

*(English)*Zbl 0691.57011The purpose of this very long paper is to prove the pseudoisotopy stability theorem in the smooth category. More precisely, if \({\mathcal C}(M)\) denotes the space of self-diffeomorphisms of \(M\times I\), which restrict to the identity on \(M\times O\cup \partial M\times I\), carrying the weak smooth topology, then the suspension map from \({\mathcal C}(M)\) to \({\mathcal C}(M\times I)\) is k-connected under (for example) the numerical conditions \(k\geq 4\) and dim \(M\geq 3k+4\). Hence in relatively low dimensions the homotopy of \({\mathcal C}(M)\) can be studied in terms of the homotopy of the infinite loop space \({\mathbb{P}}(M)=\lim_{m\to \infty}{\mathcal C}(M\times I^ m)\), itself closely related to Waldhausen’s geometric K- group A(M). I will discuss what seem to me to be limitations of these results in a final paragraph, but first I wish to give some account of the methods and results, which I hope will be intelligible to the non- specialist. Official health warning: the author has made every effort to be accurate in his details, but this area is notorious for technical problems arising in the published arguments. The main tools are adapted from the parametrized Morse theory of J. Cerf, in which one defines a Morse function on the total space of a bundle of manifolds parametrized by points of the base B, and A. Hatcher’s handle exchange theorem, which under certain conditions allows the exchange of i- for \((i+2)\)-handles. Stated in this generality, this pair of results can be combined to give a heuristic proof of the stability theorem - in the present paper, as the author makes clear, only those lemmas are proved which are necessary for the main theorem. For the sake of simplicity consider the argument for the closed disc \(D^ n.\)

The space of all submanifolds of \({\mathbb{R}}^ n\) diffeomorphic to \(D^ n\) is a model for the classifying space B\({\mathcal C}(D^{n-1})\), and may be studied via the space \(\tilde {\mathcal H}^ h({\mathbb{R}}^ n)\) of all pairs (*\(\simeq V,h)\), V being a compact contractible n-submanifold of \({\mathbb{R}}^ n\) and h: \(V\to [0,]\) a function of generalized Morse type. This last condition means that \(h^{-1}()=\partial V\) and h has only Morse and birth-death singularities (modulo local renormalization in some neighbourhood for the latter type h has the form \(y^ 3_ 1+Q(y_ 2,...,y_ n))\). The k-connectivity of the suspension for \({\mathcal C}(D^{n-1})\) is now equivalent to the k-connectivity of a compatible “suspension” from \(\tilde {\mathcal H}^ h({\mathbb{R}}^ n)\to \tilde {\mathcal H}^ h({\mathbb{R}}^{n+1})\). Assuming that \(k\sim n/3\), \(\tilde {\mathcal H}^ h\) can be replaced by its subspace \({\mathcal H}^ h_{k+2,k+3}\) consisting of pairs (*\(\simeq V,h)\), where h has one non-degenerate critical point of index 0, and all other critical points are either non-degenerate of index \(k+2\) or \(k+3\), or birth-death points of index \((=index\) Q) \(k+2\). This is a smooth version of Hatcher’s two index theorem, and receives a detailed proof in Chapter VI (80 pages). Intuitively k-connectivity of the suspension now follows because the approximating space is equivalent to the space of contractible cell-complexes consisting of \((k+2)\) and \((k+3)\)-cells attached to a single 0-cell with the attaching maps for each cell well-defined up to rotation. However the technical proof uses properties of stratified sets, which are carefully discussed in Chapter III (62 pages).

I have already mentioned that the stability theorem forms part of the programme of studying the homotopy type of the space \({\mathcal C}(M)\) by means of K-theory. This translation of problems of simple homotopy type to questions of homotopy in a classifying space is certainly elegant, but seems to offer no more than limited help in solving specific geometric problems. For example let \(C_ p\) be a cyclic group of prime order acting freely on \(S^ 3\)- then using the homotopy equivalence \(SO(4)\simeq SDiff(S^ 3)\) and the unstable theory of maps between classifying spaces, one can show that the action is homotopically equivalent to a specific free linear action. At least if the action extends to \(C_{2p}\) there is some geometric evidence that the homotopy above is actually simple, and it would be nice to be able to prove this using (say) some variant of the Waldhausen functor A( ). However any evidence provided by this method would appear at best to be stable, that is, one would not obtain the specific representation provided in the more elementary theory by \([BC_ p,BSO(4)]\). And stably the problem of characterizing allowable Reidemeister torsions is much easier - thus if the \(C_ p\) action embeds in a \(C_ pt\) action for arbitrarily large values of t, the associated Reidemeister torsion must be linear. (Intuitively this follows because we are approximating a free SO(2)- action, and these are covered by the theory of Seifert fibrations). As a mathematician interested in low-dimensional geometric problems of this kind, I am therefore left hoping that this elaborately constructed machine will eventually be sufficiently well-understood to provide real help.

The space of all submanifolds of \({\mathbb{R}}^ n\) diffeomorphic to \(D^ n\) is a model for the classifying space B\({\mathcal C}(D^{n-1})\), and may be studied via the space \(\tilde {\mathcal H}^ h({\mathbb{R}}^ n)\) of all pairs (*\(\simeq V,h)\), V being a compact contractible n-submanifold of \({\mathbb{R}}^ n\) and h: \(V\to [0,]\) a function of generalized Morse type. This last condition means that \(h^{-1}()=\partial V\) and h has only Morse and birth-death singularities (modulo local renormalization in some neighbourhood for the latter type h has the form \(y^ 3_ 1+Q(y_ 2,...,y_ n))\). The k-connectivity of the suspension for \({\mathcal C}(D^{n-1})\) is now equivalent to the k-connectivity of a compatible “suspension” from \(\tilde {\mathcal H}^ h({\mathbb{R}}^ n)\to \tilde {\mathcal H}^ h({\mathbb{R}}^{n+1})\). Assuming that \(k\sim n/3\), \(\tilde {\mathcal H}^ h\) can be replaced by its subspace \({\mathcal H}^ h_{k+2,k+3}\) consisting of pairs (*\(\simeq V,h)\), where h has one non-degenerate critical point of index 0, and all other critical points are either non-degenerate of index \(k+2\) or \(k+3\), or birth-death points of index \((=index\) Q) \(k+2\). This is a smooth version of Hatcher’s two index theorem, and receives a detailed proof in Chapter VI (80 pages). Intuitively k-connectivity of the suspension now follows because the approximating space is equivalent to the space of contractible cell-complexes consisting of \((k+2)\) and \((k+3)\)-cells attached to a single 0-cell with the attaching maps for each cell well-defined up to rotation. However the technical proof uses properties of stratified sets, which are carefully discussed in Chapter III (62 pages).

I have already mentioned that the stability theorem forms part of the programme of studying the homotopy type of the space \({\mathcal C}(M)\) by means of K-theory. This translation of problems of simple homotopy type to questions of homotopy in a classifying space is certainly elegant, but seems to offer no more than limited help in solving specific geometric problems. For example let \(C_ p\) be a cyclic group of prime order acting freely on \(S^ 3\)- then using the homotopy equivalence \(SO(4)\simeq SDiff(S^ 3)\) and the unstable theory of maps between classifying spaces, one can show that the action is homotopically equivalent to a specific free linear action. At least if the action extends to \(C_{2p}\) there is some geometric evidence that the homotopy above is actually simple, and it would be nice to be able to prove this using (say) some variant of the Waldhausen functor A( ). However any evidence provided by this method would appear at best to be stable, that is, one would not obtain the specific representation provided in the more elementary theory by \([BC_ p,BSO(4)]\). And stably the problem of characterizing allowable Reidemeister torsions is much easier - thus if the \(C_ p\) action embeds in a \(C_ pt\) action for arbitrarily large values of t, the associated Reidemeister torsion must be linear. (Intuitively this follows because we are approximating a free SO(2)- action, and these are covered by the theory of Seifert fibrations). As a mathematician interested in low-dimensional geometric problems of this kind, I am therefore left hoping that this elaborately constructed machine will eventually be sufficiently well-understood to provide real help.

Reviewer: Ch.Thomas

##### MSC:

57R50 | Differential topological aspects of diffeomorphisms |

18F25 | Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects) |

57R70 | Critical points and critical submanifolds in differential topology |

57R65 | Surgery and handlebodies |