摘要
对于任何单连通n维(n≥5)闭流形,如果不是Spin-流形,都允许有正数量曲率的黎曼度量。Spin-流形允许这样的度量当且仅当其Atiyah-Milnor不变量为0.对任意2n维quasitoric-流形π:M^(2n)→P^(n),设F(P^(n))={F_(1),…,F_(m)}是P^(n)中所有余一维面的集合,Z[F_(1),…,F_(m)]/I_(Pn)是P^(n)的面环,且λ(F_(j))=(l_(1j),…,l_(nj)),j=1,…,m是P^(n)的示性函数。令θ_(i):=li1F_(1)+…+l_(im) F_(m),1≤i≤n,J_(Pn)表示由θ_(1),…,θ_(n)生成的Z[F_(1),…,F_(m)]中的理想。关于M^(2n)的上同调环和Stiefel-Whitney类,有H∗(M^(2n),Z)=Z[F_(1),…,F_(m)]/(I_(Pn)+J_(Pn)),ω(M^(2n))=j∗Πi=1 m(1+Fi)mod 2,可知M^(2n)带有Spinc-结构,这里c=j∗Πi=1 m(1+Fi).当n=4k+2且M^(2n)是Spinc-流形时,设B是M^(2n)的一个子流形且[B]∈H8k+2(M^(2n),Z)是c的Poincaré对偶。文章利用张伟平[7]给出的Rokhlin-同余公式,计算了B的Atiyah-Milnor不变量,并给出了该不变量为0的一个充分必要条件。计算主要利用了如下结论:对于quasitoric-流形π:M^(2n)→P^(n),取P^(n)的任意顶点υ=F_(1)∩…∩F_(n),则有[F_(1)…F_(n)],[M^(2n)]=±1,其中[M^(2n)]是基本类。
It is well known that any simply connected closed n-manifold of dimension n≥5 admits a Riemannian metric with positive scalar curvature if it is not Spin,and a Spin manifold admits such a metric if and only if its Atiyah-Milnor invariant vanishes.For any 2n-dimensional quasitoric manifoldπ:M^(2n)→P^(n)let F(P^(n))={F_(1),…,F_(m)}be the set of all the codimension surfaces in P^(n),Z[F_(1),…,F_(m)]/I_(Pn)be the Torus of P^(n),andλ(F_(j))=(l1j,…,l_(nj)),j=1,…,m be the characteristic function of P^(n).Letθ_(i):=li1F_(1)+…+l_(im) F_(m),1≤i≤n,and J_(Pn)denote ideals in Z[F_(1),…,F_(m)]generated byθ_(1),…,θ_(n).For the cohomology rings of M^(2n)and Stiefel-Whitney classes,we have H∗(M^(2n),Z)=Z[F_(1),…,F_(m)]/(I_(Pn)+J_(Pn)),ω(M^(2n))=j∗Πi=1 m(1+Fi)mod 2,We know that M^(2n)has a spinc-structure,wherec=j∗Πi=1 m(1+Fi).If n=4k+2 and M^(2n)is a Spin-manifold,let B be a submanifold of M^(2n)and B in H8k+2(M^(2n),Z),be a Poincaréduality of c.In this paper,it is calculated that the Atiyah-Milnor invariant of B and given a necessary and sufficient condition to vanish of this invariant using the remarkable Rokhlin-congruence formula found by Zhang Weiping[7].Calculations are based on the following conclusions:for quasitoric manifoldπ:M^(2n)→P^(n)letυ=F_(1)∩…∩F_(n),then[F_(1)…F_(n)],[M^(2n)]=±1,where[M^(2n)]is the fundamental class.
作者
叶蔚聪
刘昌莲
YE Wei-cong;LIU Chang-lian(College of Mathematics and Statistics,Guangxi Normal University,Guilin,Guangxi,541006,China)
出处
《新疆师范大学学报(自然科学版)》
2023年第4期34-42,共9页
Journal of Xinjiang Normal University(Natural Sciences Edition)