The averaging theorem of Lefschetz coincidence numbers and estimation of Nielsen coincidence numbers

Bull. Korean Math. Soc. 1996 Vol. 33, No. 2, 205-211

Chan Gyu Jang Chonbuk National University

Abstract : A well-known averaging theorem of Lefschetz number of a self map $f \: X \rightarrow X$ is a relation between the Lefschetz number of $f$ and the Lefschetz numbers of lifts of f. That is, if $f\:X \rightarrow X$, $K$ is a normal subgroup of $\pi_1(X)$ with $f_\sharp(K)\subset K$, and $\pi_1(X)\slash K$ is finite, then $$ L(f)=\frac{1}{[\pi_1(X)\: K]}\Sigma_{\tilde f}L(\tilde f), $$ the summation is over all lifts $\tilde f$ of $f$ ([2], Chapter 3,2.12 Theorem). Fixed-point theory has a natural extension to coincidences (Theorem 2.7). C.K.McCord [4] referred the averaging theorem in coincidence theory. In this paper, we give alternative proof of the averaging theorem in coincidence theory by the same way as proved the averaging theorem in fixed theory, and give an estimatin of coincidence Nielsen number which is also an extension of an estimation of Nielsen number in Fixed-point theory(Theorem 2.10). We will briefly review the coinidence theory in section 1 and covering theory in section 2 on the basis of the developement of [4].