On pseudo-isotopy classes of homeomorphisms of \(\#_p(\mathbb{S}^1\times \mathbb{S}^n)\)
Let \(Y\) be the \((n+2)\)-dimensional handlebody obtained by attaching \(p\) orientable 1-handles to the \((n+2)\)-ball (\(n\geq 3\), \(p\geq 1\)) and let \(X\) be the boundary of \(Y\), which is the connected sum of (\(S^{n+1}\) and) \(p\) copies of \(S^1\times S^n\). In the present paper the authors investigate the group \({\mathcal D}(X)\) of pseudo-isotopy (or concordance) classes of automorphisms of \(X\), in either the smooth or the PL category, and extend to higher \(n\) the results obtained for the case \(n=2\) by \textit{F. Laudenbach} and \textit{V. Poénaru} [Bull. Soc. Math. France 100, 337-344 (1972; Zbl 0242.57015)] and \textit{J. M. Montesinos} [Geometric topology, Proc. Conf., Athens/Ga. 1977, 219-237 (1979; Zbl 0479.57007)] and the results valid for \(p=1\), contained in the papers by \textit{H. Gluck} [Bull. Am. Math. Soc. 67, 586-589 (1961; Zbl 0173.51402)], \textit{W. Browder} [Trans. Am. Math. Soc. 128, 155-163 (1967; Zbl 0166.19702)], and \textit{R. K. Lashof} and \textit{J. L. Shaneson} [Bull. Am. Math. Soc. 75, 171-175 (1969; Zbl 0198.28701)]. Building on their previous results [Topology Appl. 53, No. 2, 111-118 (1993; Zbl 0820.57018); Forum Math. 6, No. 4, 415-429 (1994; Zbl 0822.57015)], the authors prove that the natural map from \({\mathcal D}(X)\) to the group \({\mathcal E}(X)\) of homotopy classes of homotopy self-equivalences of \(X\) is an isomorphism. \({\mathcal D}(X)\) is proved to be generated by a reflection and a finite set of automorphisms of \(X\) (and \(Y\)) which permute the handles, slide a handle over another, switch the two ends of a handle, or twist a handle along its belt sphere. Consequently every automorphism of \(X\) extends to one of \(Y\). It also follows that the natural homomorphism \({\mathcal D}(X)\to\text{Out}(\pi_1X)\) is surjective, and the authors prove that its kernel is generated by a suitably chosen reflection and the twists of the handles (which they call handle rotations) and is thus isomorphic to the direct sum of \(p+1\) copies of \({\mathbb Z}_2\).
group of homotopy classes of homotopy equivalences
scientific article; zbMATH DE number 1187540