Manifolds compatible with the point with respect to involutions
The authors introduce the following notion of Property CP (\textit{compatible with the point}): A compact smooth manifold \(F\) is said to have Property CP (``compatible with a point'') if there is a (smooth) involution \((M,T)\) (that is, \(T:M\to M\) is a smooth involution on a compact of a smooth compact manifold \(M\),) such that the fixed point set of \(T\) is a disjoint union \(F\sqcup \{p\}\) where \(p\) is a point. It was shown by Conner and Floyd in 1964 that the only spheres which have the property CP are \(\mathbb S^n, n=1,2,4,8\), and \(\dim M=2\dim F=2n.\) Extending this result, Pergher classified products of spheres which have property CP. The real projective space \(\mathbb RP^n\) has the property CP, as can be seen by considering the involution \(T: \mathbb RP^{n+1}\to \mathbb RP^{n+1}\) induced by the reflection about the hyperplane \(\mathbb R^{n+1}\subset \mathbb R^{n+1}\). The same argument also shows that \(\mathbb CP^n\) and \(\mathbb HP^n\) have the property CP. The property CP is not an invariant of the unoriented cobordism. It is also not preserved by taking products. The authors consider class of Dold manifolds \(P(m,n)\), which is the quotient of \(\mathbb S^m\times \mathbb CP^n\) by the involution \((v,[z_0,\ldots,z_n])\mapsto (-v,[\bar z_0,\ldots, \bar z_n])\). Their main results, proved in Theorems 3.1-3.4, are the following: Let \(m,n\ge 1\). Then \(P(m,n)\) does not have property CP if any of the following hold: (1) \(m\) is odd, (2) \(m\equiv 0 \pmod 4, n,\) odd, (3) \(m=2, n,\) even, and, (4) \(6<m\le n\), and \(n\) odd. A crucial ingredient needed in the proofs are the Conner-Floyd theory, [\textit{P. E. Conner} and \textit{E. E. Floyd}, Differentiable periodic maps. Springer-Verlag, Berlin (1964; Zbl 0125.40103)] especifically, relation between the \(\mathbb Z_2\)-equivariant cobordism of an involution \((M,T)\) and its fixed point data, and the exactness of the Conner-Floyd exact sequence \(0\to \mathcal N_m^{\mathbb Z_2}(*)\to \bigoplus_{0\le j\le m} \mathcal N_j(BO(m-j))\to \mathcal N_{m-1}(BO(1)) \to 0\). A result of [\textit{D. C. Royster}, Indiana Univ. Math. J. 29, 267--276 (1980; Zbl 0406.57027)] is used to show, in Theorem 2.2, that if \(F\) is not null-cobordant and if \(\dim F\) is odd, then \(F\) does not have property CP. It is deduced from this that, if \(\dim F=n\) is odd, and if \(x^n[N]=0\) for all \(x\in H^1(F;\mathbb Z_2)\), then \(F\) does not have property CP. The authors use this to conclude that \(P(m,n)\) does not have property CP if \(m\) is odd. Proofs of tatements (2), (3), and (4) are much more involved. Among other things, a result of [\textit{R. E. Stong}, Fundam. Math. 169, No. 1, 85--95 (2001; Zbl 0976.55013)] on the classification of total Stiefel-Whitney classes of vector bundles over the Dold manifolds \(P(m,n)\), is crucially used. The authors introduce a certain characteristic class, \(\widetilde{W}(\lambda)\) for a line bundle \(\lambda\) over a smooth manifold \(X\) as the total Stiefel-Whitney class of the bundle \(\tau X\otimes \lambda\oplus \lambda\), where \(\tau X\) denotes the tangent bundle of \(X\). The authors obtain show that \(\widetilde{W}(\lambda)=W(\pi^*(\tau F)\otimes \lambda\oplus \lambda)\) is obtained when \(\lambda\) is the the Hopf bundle over \(\mathbb RP(\eta)\) associated to a projective space bundle \(\pi: \mathbb RP(\eta)\to F\). This is applied to the situation when \(P(m,n)\sqcup \{point\}\) is the fixed set of an involution \((M,T),\) taking \(\eta\to P(m,n)\) to be the normal bundle to the inclusion \(P(m,n)\hookrightarrow M\). Using Stong's results, the formula for \(\widetilde{W}(\lambda)\), and some intricate computations involving characteristic classes, enable the authors to settle the remaining three cases, (2), (3), and (4). The authors also obtain some general results concerning property CP. We mention two such results. Theorem 2.3: Suppose that \(V^m,W^n\) are such that their dimensions have non-disjoint dyadic expansions, i.e., \(m=\sum m_k2^k,n=\sum n_k2^k\), \(m_k,n_k\in \{0,1\}\) and \(m_r=1=n_r\) for some \(r\ge 0\). If \(m+n\) is odd, then \(V\times W\) does not have property CP. Theorem 2.4: If \(F\) has property CP, then so does the cartesian product \(F^{2^s}\) for any \(s\ge 1\).
scientific article; zbMATH DE number 7634038
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