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**Fixed point sets and tangent bundles of actions on disks and Euclidean spaces.**
*(English)*
Zbl 0861.57047

The main result of the paper is the determination, for any group not of prime power order, of exactly which smooth manifolds can be the fixed point set of smooth \(G\)-actions on disks or on Euclidean spaces. General techniques for constructing actions on disks with a fixed point set of a given homotopy type were developed in [the author, Comment. Math. Helv. 50, 155-177 (1975; Zbl 0304.57020)], and the procedure for constructing actions on Euclidean spaces is similar, but simpler. What is new in the paper is the way of constructing a \(G\)-vector bundle over a \(G\)-complex of a given homotopy type which extends a given \(G\)-bundle over the fixed point set. Such a \(G\)-bundle can then be used to control the process of equivariantly thickening up the \(G\)-complex [A. L. Edmonds and R. Lee, Topology 14, 339-345 (1975; Zbl 0317.57018) and K. Pawałowski, ibid. 28, 273-289 (1989; Zbl 0691.57017)] to get a manifold with a smooth \(G\) action having a given homotopy type, and in particular to control the diffeomorphism type of the fixed point set. In previous work by various authors imposed additional assumptions on the bundles resulting in less general (incomplete) results.

There is an integer \(n_G\geq 0\), called the Oliver number, with the property that a finite CW complex \(F\) is homotopy equivalent to the fixed point set of some smooth \(G\) action on a disk if and only if \(\chi(F)\equiv 1\pmod{n_G}\).

Theorem 1. Let \(G\) be any finite group not of prime power order. Fix a smooth manifold \(F\) and a \(G\)-vector bundle \(\eta\downarrow F\) satisfying the following three conditions: (1) \(\eta\) is nonequivariantly a product bundle; (2) for each prime \(p\) dividing the order of \(G\) and each \(p\)-subgroup \(P\subseteq G\), \([\eta|P]\) is infinitely \(p\)-divisible in \(\widetilde{KO}_P(F)_{(p)}\), where \(\widetilde{KO}_P(F)= KO_P(F)/KO_P(pt)\); and (3) \(\eta^G\cong\tau(F)\), the tangent bundle of \(F\). Then there is a smooth action of \(G\) on a contractible manifold \(M\) such that \(M^G=F\), and such that \(\tau(M)|F\cong\eta\oplus(V\times F)\) for some representation \(V\) with \(V^G=0\). If \(\partial F=\emptyset\), then \(M\) can be chosen to be a Euclidean space. If \(F\) is compact and \(\chi(F)\equiv 1\pmod{n_G}\), then \(M\) can be chosen to be a disk.

Conditions (1)–(3) in the theorem are also necessary. Let \({\mathfrak F}{\mathfrak i}{\mathfrak x}(G)\) be the class of smooth manifolds \(F\) for which there is a \(G\)-bundle \(\eta\downarrow F\) satisfying conditions (1)–(3) in the theorem. A further analysis of \({\mathfrak F}{\mathfrak i}{\mathfrak x}(G)\) depends on representation theory. Let \({\mathcal M}_{\mathbb{C}}\supseteq {\mathcal M}_{\mathbb{C}^+}\supseteq{\mathcal M}_{\mathbb{R}}\) be the classes of finite groups for which there exist representations \(V\) and \(W\) which are complex, self-conjugate, or real, respectively, such that \(V|P\approx W|P\) for any \(P\subseteq G\) of prime power order, and such that \(\dim(V^G)=1\) and \(\dim(W^G)=0\). There are also group theoretic characterizations for these groups, see Lemma 3.1 of the paper. Depending on whether the Sylow-2 subgroup of \(G\) is normal in \(G\) or not and whether \(G\in{\mathcal M}_{\mathbb{R}}\), \(G\in{\mathcal M}_{\mathbb{C}^+}\smallsetminus{\mathcal M}_{\mathbb{R}}\), \(G\in{\mathcal M}_{\mathbb{C}}\smallsetminus{\mathcal M}_{\mathbb{C}^+}\), or \(G\not\in{\mathcal M}_{\mathbb{C}}\), the author gives \(K\)-theoretic conditions on the class of \(\tau(F)\) which are necessary and sufficient for \(F\) to be in \({\mathfrak F}{\mathfrak i}{\mathfrak x}(G)\).

The paper has an appendix which makes the paper much more readable. Here, the author collects results which are well-known, but which are either hard to find in the literature, or which are used often enough in the paper to state them explicitly.

There is an integer \(n_G\geq 0\), called the Oliver number, with the property that a finite CW complex \(F\) is homotopy equivalent to the fixed point set of some smooth \(G\) action on a disk if and only if \(\chi(F)\equiv 1\pmod{n_G}\).

Theorem 1. Let \(G\) be any finite group not of prime power order. Fix a smooth manifold \(F\) and a \(G\)-vector bundle \(\eta\downarrow F\) satisfying the following three conditions: (1) \(\eta\) is nonequivariantly a product bundle; (2) for each prime \(p\) dividing the order of \(G\) and each \(p\)-subgroup \(P\subseteq G\), \([\eta|P]\) is infinitely \(p\)-divisible in \(\widetilde{KO}_P(F)_{(p)}\), where \(\widetilde{KO}_P(F)= KO_P(F)/KO_P(pt)\); and (3) \(\eta^G\cong\tau(F)\), the tangent bundle of \(F\). Then there is a smooth action of \(G\) on a contractible manifold \(M\) such that \(M^G=F\), and such that \(\tau(M)|F\cong\eta\oplus(V\times F)\) for some representation \(V\) with \(V^G=0\). If \(\partial F=\emptyset\), then \(M\) can be chosen to be a Euclidean space. If \(F\) is compact and \(\chi(F)\equiv 1\pmod{n_G}\), then \(M\) can be chosen to be a disk.

Conditions (1)–(3) in the theorem are also necessary. Let \({\mathfrak F}{\mathfrak i}{\mathfrak x}(G)\) be the class of smooth manifolds \(F\) for which there is a \(G\)-bundle \(\eta\downarrow F\) satisfying conditions (1)–(3) in the theorem. A further analysis of \({\mathfrak F}{\mathfrak i}{\mathfrak x}(G)\) depends on representation theory. Let \({\mathcal M}_{\mathbb{C}}\supseteq {\mathcal M}_{\mathbb{C}^+}\supseteq{\mathcal M}_{\mathbb{R}}\) be the classes of finite groups for which there exist representations \(V\) and \(W\) which are complex, self-conjugate, or real, respectively, such that \(V|P\approx W|P\) for any \(P\subseteq G\) of prime power order, and such that \(\dim(V^G)=1\) and \(\dim(W^G)=0\). There are also group theoretic characterizations for these groups, see Lemma 3.1 of the paper. Depending on whether the Sylow-2 subgroup of \(G\) is normal in \(G\) or not and whether \(G\in{\mathcal M}_{\mathbb{R}}\), \(G\in{\mathcal M}_{\mathbb{C}^+}\smallsetminus{\mathcal M}_{\mathbb{R}}\), \(G\in{\mathcal M}_{\mathbb{C}}\smallsetminus{\mathcal M}_{\mathbb{C}^+}\), or \(G\not\in{\mathcal M}_{\mathbb{C}}\), the author gives \(K\)-theoretic conditions on the class of \(\tau(F)\) which are necessary and sufficient for \(F\) to be in \({\mathfrak F}{\mathfrak i}{\mathfrak x}(G)\).

The paper has an appendix which makes the paper much more readable. Here, the author collects results which are well-known, but which are either hard to find in the literature, or which are used often enough in the paper to state them explicitly.

Reviewer: K.H.Dovermann (Honolulu)