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ABSTRACT. The notion of coherency with submanifolds for a Morse function on a manifold is introduced and discussed in a general way. A Morse inequality for a given periodic transformation which compares the invariants called qlh Euler numbers on fixed point set and the invariants called 17th Lefschetz numbers of the transformations is thus obtained. This gives a fixed point theorem in terms of qth Lefschetz number for arbitrary q.Let /be a periodic transformation of a closed m-dimensional manifold M with fixed point set N. We develop in this note an equivariant approach using Morse theory. We introduce in §2 the notion of coherency with a submanifold S of M for a Morse function and show that such S-coherent Morse functions are dense in CX(M). Furthermore, in this approximation/-invariance will be preserved ( §3). The coherency with the fixed point set N of/makes it possible to compare the difference of qth Euler number of N and qth Lefschetz number of /. More precisely, let ßq(N) and Xq(f) be respectively the ^th Betti numbers of N and the trace of/* on the qth homology group Hq(M) with real coefficients. Let Bq(N) and Aq(f) be their alternative sums respectively, i.e.,

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