A gap theorem for Lusternik–Schnirelmann π1-category

✍ Scribed by Erkki Laitinen; Takao Matumoto

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 83 KB
• Volume: 93
• Category: Article
• ISSN: 0166-8641
• DOI: 10.1016/s0166-8641(97)00259-9

✦ Synopsis

The Lusternik-Schnirelmann π1-category, catπ 1 X, of a topological space X is the least integer n such that X can be covered by n + 1 open subsets U0, . . . , Un, every loop in each of which is contractible in X. In this paper we will prove a gap theorem that catπ 1 M n = n -1 for any closed connected n-dimensional manifold M n . With the fact that the fundamental group of a compact Kähler manifold is not a nontrivial free group, we see as a corollary that the π1-category of a compact Kähler surface is even.