On minimal fixed point numbers of relative maps

✍ Scribed by Xuezhi Zhao

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 303 KB
• Volume: 112
• Category: Article
• ISSN: 0166-8641
• DOI: 10.1016/s0166-8641(99)00224-2

✦ Synopsis

Let f : (X, A) → (X, A) be a self map of a pair of compact polyhedra. We define two new Nielsen type numbers m(f ; X, A) and m(f ; X -A), which are lower bounds for the number of fixed points on X and on Cl(X -A), the closure of X -A in X, respectively. These relative homotopy theoretic lower bounds can be realized without the by now familiar by-passing condition.

Part of our intention is that m(f ; X, A) and m(f ; X -A) be companion numbers to the surplus number SN(f ; X -A) in the sense that, as much as possible, these three numbers should be realizable simultaneously. This is analogous to the realizability of the numbers N(f ; X, A), N(f ; X, A) and N(f ; X -A) which, in the presence of by-passing (together with the usual Wecken type conditions), can be realized simultaneously. Thus we answer open questions posed by Schirmer.