Celebratio Mathematica — Freedman — Topological Poincaré

The topological Poincaré conjecture in dimension 4 (the work of M. H. Freedman)

Filter the Bibliography List

year	title	people

			clear

article M. H. Freedman: “A fake \( S^{3}\times \mathbf{R} \),” Ann. of Math. (2) 110 : 1 (1979), pp. 177–201. MR 0541336 Zbl 0442.57014

This paper describes a construction which can be exploited to recover the results of higher dimensional surgery (\( \dim \geq 5 \)) for a wide variety of noncompact four dimensional surgery problems. These include any countable union over boundary components of compact problems with simply connected targets. Here we will concentrate on a special case: the construction of a 4-manifold \( W \) proper-homotopy-equivalent to \( \mathbb{S}^3\times\mathbb{R} \) which has a cross section diffeomorphic to the Poincaré sphere \( \Sigma^3 \) and thus cannot be diffeomorphic to \( \mathbb{S}^3\times\mathbb{R} \). Although the ideas of surgery are in the background, our arguments do not rely heavily on them. A sequel will appear to develop a version of the surgery exact sequence for noncompact 4-manifolds.

F. Ancel: “Nowhere dense tame 0-dimensional decompositions of \( {S}^4 \) (an exposition of a theorem of Mike Freedman)”. Unpublished manuscript, 1981. This work was originally (incorrectly) cited by Siebenmann in his list of references as “Nowhere dense tame 0-dimensional decompositions of \( S^4 \); an exposition of a theorem of Mike Freedman”. Ric Ancel notes the correct title (email to the editor, dated 28 May 2020) and has made the PDF available to readers of CM. techreport

L. Siebenmann: “La conjecture de Poincaré topologique en dimension 4 (d’après M. H. Freedman),” pp. 219–248 in Bourbaki Seminar, vol. 1981/1982. Astérisque 92. Soc. Math. France, Paris, 1982. MR 689532 incollection