Equivariant Intersection Theory

Let f be a smooth self-map of a compact smooth manifold M. We may ask if f is homotopic to a fixed point free map. Wecken(1942) and Jiang(1981) showed that if dimension of M is bigger than 2 then there is an invariant in a framed bordism group which vanishes if and only if f is homotopic to a fixed point free map. The next question we may ask if this invariant is not trivial, what can we say about the fixed point set? In this work, we will show that we can extract important information from this invariant by using the geometric point of view. The proof of the main theorem relies heavily on intersection theory. The key ingredients in the proof are fact about transversality, embedding theory and classifying spaces which in general do not hold for the equivariant setting. Fortunately, by using results of Petrie on equivariant transversality, Komiya on equivariant embedding, we are able to prove the main theorem for the equivariant case.