Bordism groups and shape theory (Q802169)

Let G be one of the classical groups O, U, SO, SU, Spin, Sp. With each of these groups are associated bordism functors \(\Omega^ G_ n\), \(n\in {\mathbb{Z}}^+\) and the Thom spectrum \(\{TBG_ n\}\) [see, e.g., \textit{R. E. Stong}, Notes on cobordism theory (1968; Zbl 0181.266)]. The elements of \(\Omega^ G_ n(X)\) are classes of cobordant singular G-manifolds \(f: M^ n\to X\) of the space X. The author first considers compact metric spaces X and replaces maps f by strong shape morphisms \(M^ n\to X\) [in the sense of the author, Pac. J. Math. 6, 25-65 (1976; Zbl 0346.55014)], obtaining thus new functors \({\bar \Omega}^ G_ n\). Moreover, he defines homology groups \(\overline{TBG}_ n(X)\) as \(\text{colim}_ k {\bar \pi}_{n+k}(TBG_ k\overline\bigwedge X)\), where \({\bar \pi}_ m\) denotes the m-th strong shape group. The main result asserts that for strongly shape connected X there exists a natural isomorphism \({\bar \Omega}^ G_ n(X)\approx \overline{TBG}_ n(X)\), \(n>0\). In the case of CW-complexes X, this isomorphism reduces to the well-known Thom isomorphism. Using compact supports, the author extends his result also to the non-compact case.