Celebratio Mathematica — Freedman

Topology in dImension 4

Works connected to Laurence Robert Taylor

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article M. H. Freedman and L. Taylor: “\( \Lambda \)-splitting 4-manifolds,” Topology 16 : 2 (1977), pp. 181–184. MR 0442954 Zbl 0363.57004

article M. H. Freedman and L. R. Taylor: “A universal smoothing of four-space,” J. Differential Geom. 24 : 1 (1986), pp. 69–78. MR 0857376 Zbl 0586.57007

Except in dimension four, smooth structures can be classified up to \( \varepsilon \)-isotopy by bundle reductions. Since \( \mathbb{R}^n \) is contractible, this implies that any smooth structure \( \Gamma \) on \( \mathbb{R}^n \), \( n \neq 4 \), is \( \varepsilon \)-isotopic to the standard one. In contrast, \( \mathbb{R}^4 \) has many distinct smoothings (even up to diffeomorphism.) We construct a certain smoothing of the half-space, \[ \tfrac{1}{2}\mathbb{R}^4 = \{(x_1,x_2,x_3,x_4)|x_4\geq 0\} \] and write \( H \) for this half-space together with its smooth structure. \( H \) contains all other smoothings of \( \frac{1}{2}\mathbb{R}^4 \) and is unique with respect to this property. \( H \) is the universal half-space. The interior \( \mathring H = U \) is naturally identified (replace \( x_4 \) with \( \ln x_4 \)) with a smoothing of \( \mathbb{R}^4 \). Corollary B states that \( U \) contains every smoothing of \( \mathbb{R}^4 \) imbedded within it. Thus, we say \( U \) is a universal \( \mathbb{R}^4 \). The construction of \( U \) is unambiguous but we do not claim that any \( \mathbb{R}_{\Gamma}^4 \) into which all smooth \( \mathbb{R}^4 \)’s imbed is diffeomorphic to \( U \). This is not known.