Concerning real valued maps of the š¯‘›-sphere
G. R. Livesay Ā· Proceedings of the American Mathematical Society Ā· 1957
A definition for the width of a closed curve, believed to be new, is used together with some standard homology theory, to prove the following theorem: Let f: Sn-*EI be continuous, 0 < d < 2, p the covering map Sn -pn. If Xd= X{XESnI there exists y GSn with p(x, y) =d, f(x) =f(y) }, then PXd carries the nontrivial mod 2 Cech n -1 cycle of pn. This generalizes Theorem 2 of [4], already considerably extended in other directions by Bourgin [2] and Yang [5]. We will use the following notation: En+, = Euclidean n +1 space, p is the Euclidean metric, co is the origin in En+,. Sn = {xGEn+, p(x, ) =1}. pn is projective n space. pi:A,XA2-*Ai, i=1, 2 will denote the projection. fCp(A)=p dimensional Cech homology group of A with coefficients the integers mod 2 (the only coefficients to be used here). T2=S'XS', A=diagonal in T2. In a space X, A =X-A,A = closure of A.