A New Approach to the Critical Value Theory

These formulas hold for any topological group-system 2, any subsystem Si of 2 , and the difference-system S —Si. (W. Mayer, Topologische Gruppensysterne, Monatshefte für Mathematik und Physik, vol. 47 (1938); henceforth referred to as M, TG.) Here 5<(S) denotes the i-dimensional Betti group of S, while ^»(Si) and 5t-(S —Si) are these groups for Si and S —Si respectively. The symbol r( ), of course, stands for the rank of the group in the parentheses. By Z>t(Si, S) we mean the subgroup of JB»(2I) containing all the classes of this group whose elements bound in S. The formula (I) was first derived for the case of a complex in Lefschetz' Topology, 1930 (p. 150), and independently for the complex modulo 2 by J. Rybarz, Monatshefte für Mathematik und Physik (1931). In the generality needed here the proof of (I) is given in M, TG (pp. 54-57), under the assumption, of course, that all the ranks appearing in (I) are finite, since otherwise the formula would be meaningless. But the proof there given shows also that (a) r £ * ( S S i ) = oo implies that either rJB,-(2) or r£>t_i(2i, S), or both, are infinite; (b) rBi(2i — Si) finite implies rZ)t_i(Si, S) finite, and if in addition rBi(Xi) is finite then rB{(2) is finite too; and (c) r J 3 < ( 2 Z i ) = 0 implies rDM(Hu S ) = 0 and if in addition rjB<(2i) is finite, then rB<(2i) =rJ?<(2)+rZ?<(2i, S) . As an immediate consequence of equations (I) we notice the inequality