PROCEEDINGSOF THE AMERICAN MATHEMATICAL SOCIETY Volume 115, Number 2, June 1992 ON CHANGING FIXED POINTS AND COINCIDENCES TO ROOTS ROBIN BROOKS AND PETER WONG (Communicated by Frederick R. Cohen) Abstract. The coincidence problem, finding solutions to f(x) = g(x), can sometimes be converted to a root problem, finding solutions to a(x) = a . As an application, we show that for any two maps f, g: M -* M, N(f, g) = \L(f, g)\ where M is a compact connected nilmanifold, N(f, g) and L(f, g) are the Nielsen and Lefschetz numbers, respectively, of / and g . This result in the case where g is the identity is due to D. Anosov. 1. Introduction Let G be a compact connected Lie group and AT be a closed subgroup. Denote by Af = G/K the homogeneous space of right cosets. In [7] Fadell observed that every selfmap /: Af —> Af has a fixed point if and only if there is a solution to the root problem ipig) = eK (¿? e G the unit) for every AT-map \p: G -* Af. Here AT acts on G via k o g = gk~x and AT acts on Af via k o gK = kgK. The root problem is often easier to analyze, solve [2-4, 11] in particular, because root classes of maps into closed orientable manifolds always have the same index. Accordingly, we convert the fixed point and coincidence problems for maps of nilmanifolds into a root problem and generalize to coin- cidences (Theorem 3.3) Anosov's result [1] that for any selfmap /: Af —► Af on a compact nilmanifold Af, the Nielsen fixed point classes of / have the same index each of which is 0, +1, or -1. Throughout //* and H* will denote singular homology and cohomology with integer coefficients, respectively. During the preparation of the manuscript, we learned that Theorem 3.3 was also obtained by Jezierski [9] and McCord [12] using different methods. We thank Bob Brown for bringing [9] to our attention and Chris McCord for his preprint. We also thank the referee for a number of helpful suggestions. 2. Local Nielsen root theory In this section we introduce a local version of the root theory as in [2] (see also [3, 4, 11]), which we need in §3. Received by the editors May 8, 1990. 1991 Mathematics Subject Classification. Primary 55M20. Key words and phrases. Fixed points, coincidences, roots, Lefschetz number, Nielsen number, degree, nilmanifold. ©1992 American Mathematical Society 0002-9939/92 $1.00+ $.25 per page 527 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use