Classifying seven dimensional manifolds of fixed cohomology type

详细信息    查看全文

• 作者：Christine M. Escher^a ; ^{tine@math.orst.edu" class="auth_mail" title="E-mail the corresponding author} ; Pongdate Montagantirud^b ; ^{pongdate.m@chula.ac.th" class="auth_mail" title="E-mail the corresponding author}
• 关键词：55R15 ; 55R40 ; 57R20
• 刊名：Differential Geometry and its Applications
• 出版年：2016
• 出版时间：December 2016
• 年：2016
• 卷：49
• 期：Complete
• 页码：312-325
• 全文大小：375 K

文摘

We study seven dimensional manifolds of fixed cohomology type with integer coefficients: H⁰≅H²≅H⁵≅H⁷≅Z$H^0\cong H^2\cong H^5\cong H^7\cong \mathbb{Z}$, H⁴≅Z_r$H^4\cong {\mathbb{Z}}_r$, H¹=H³=H⁶=0$H^1=H^3=H^6=0$, simply called manifolds of type r  , where Z_r${\mathbb{Z}}_r$, is a cyclic group of order r   generated by the square of the generator of H²$H^2$. Such manifolds include the Eschenburg spaces, the Witten manifolds and the generalized Witten manifolds. Most spaces from these three families admit a Riemannian metric of positive sectional curvature or an Einstein metric of positive Ricci curvature. In 1991 M. Kreck and S. Stolz introduced three invariants to classify manifolds M of type r up to homeomorphism and diffeomorphism. In this article, we show that for spin manifolds of type r we can replace two of the homeomorphism invariants by the first Pontrjagin class and the self-linking number of the manifolds. This replacement has been stated without proof and been used for calculations for the Eschenburg spaces, hence our result closes a gap in the literature.