Cohomology complex projective space with degree one codimension-two fixed submanifolds
Let \(M^{2n}\) be a smooth, closed, orientable \(2n\)-manifold such that there is a class \(x\in H^2 (M;\mathbb{Z})\) with the property that \(H^* (M;\mathbb{Z}) =\mathbb{Z} [x]/(x^{n+1})\) (i.e. a cohomology complex projective \(n\)-space), \(i:K^{2n-2} \subset M^{2n}\) the inclusion map of a closed, connected, orientable submanifold and \(d\) an integer. Then \(d\) is the degree of \(K^{2n-2}\) if \(i_*[K]\) is the Poincaré dual of \(dx\) (the orientation of \(K^{2n-2}\) is chosen in such a way that \(d\) is nonnegative). Let \(D_p (M^{2n})\) \((p\) a prime number) be the set of positive integers \(d\) such that \(M^{2n}\) admits a smooth action of the cyclic group \(Gp\) of order \(p\) such that the fixed point set of the action contains a submanifold of codimension 2 of degree \(p\). The authors motivated by the work of several investigators establish the following conjecture: ``If \(D_p (M^{2n})\) is nonempty, then \(D_p (M^{2n}) =\{1\}\)''. Concerned with this conjecture the authors prove: (1) if \(p=2\) and \(l\in D_2 (M^{2n})\), then \(D_2 (M^{2n}) =\{1\}\), (2) if \(n\) is odd and \(p\) and \(q\) are primes, then \(l\in D_p (M^{2n})\) implies that either \(D_q (M^{2n})\) is empty or \(D_q (M^{2n}) =\{1\}\), (3) if \(n\) is odd or \(p=2\), then \(D_p (M^{2n})\) is a finite subset of the odd natural numbers, (4) if \(M^{4m}\) is a homotopy \(CP^{2m}\) and \(m \not\equiv 0,4\) or \(7\pmod 8\) and \(l\in D_3 (M^{4m})\), then \(D_3 (M^{4m}) =\{1\}\), (5) if \(m \not\equiv 0,4\), or \(7\pmod 8\), then \(D_3(CP^{2m}) =\{1\}\). Moreover, the paper contains similar results about the case \(p\) odd and \(n\) even when the set \(D_p (M^{2n})\) is extended to a new set \(DE_p (M^{2n})\) which gives information about the tangent representation at the isolated fixed point of the action of \(G_p\) on \(M^{2n}\) fixing a submanifold of codimension two and an isolated point.
scientific article; zbMATH DE number 940732
scientific article