Floer homology and homotopy planes (Q1384605)

Let \(P(a,b,m)\) be a homotopy plane, where \((a,b,m)\) are relatively prime integers, \(1<a<b\) and \(m>0\), parametrizing the set of algebraic isomorphism classes of this plane. A closed regular neighbourhood of a compactification divisor in a suitable algebraic compactification of \(P(a,b,m)\) is bounded by a closed homology 3-sphere \(S(a,b,m)\), whose diffeomorphism type is uniquely determined by \(P(a,b,m)\). The main theorem of this paper is: The Floer group \(HF(S(a,b,m))\) of the homology 3-sphere \(S(a,b,m)\) is a free abelian group of rank \({1\over 3} m(a^2-1)(b^2-1)\), lying in the even part of the grading. It is 4-periodic. If \(a,b\) are odd it is 2-periodic.