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Smith theory says that the fixed point of a semi-free action of a group $G$on a contractible space is ${\bb Z}_p$-acyclic for any prime factor $p$ of $G$.Jones proved the converse of Smith theory for the case $G$ is a cyclic groupacting on finite CW-complexes. We extend the theory to semi-free group actionon finite CW-complexes of given homotopy type, in various settings. Inparticular, the converse of Smith theory holds if and only if certain$K$-theoretical obstruction vanishes. We also give some examples that show theeffects of different types of the $K$-theoretical obstruction.

Fixed Point Sets and the Fundamental Group I: Semi-free Actions on G-CW-Complexes