In this article, the authors prove a relative variant of the Atiyah-Bott-Lefschetz fixed point theorem.
More precisely (but for technical details cf. Section 2 in the paper), let \((M,X)\) be a pair consisting of a smooth closed manifold \(M\) and a smooth closed submanifold \(X\subset M\). On the one hand one can introduce the concept of a relative elliptic complex over \((M,X)\) in the way analogous to the absolute case. Beyond the relative elliptic operators in this complex, one can talk about relative geometric operators too, akin to the absolute case. We demand that the elliptic operators and the geometric operators in this complex be compatible with each other in the usual sense i.e. they should satisfy the chain property. On the other hand we assume that \((M,X)\) carries a relative action of a group \(G\) as well i.e. every \(g\in G\) gives rise to a diffeomorphism \(g:M\rightarrow M\) such that \(g(X)=X\); a point \(x\in M\) is called a fixed point of \(g\) if \(g(x)=x\) and this fixed point is called non-degenerate if the spectrum of the derivative \(g_*: T_xM\rightarrow T_{g(x)}M=T_xM\) does not contain the unit \(1\in\mathrm{C}\). Note that non-degenerate fixed points are isolated hence by compactness of \(M\) there exists only finitely many of them. The main result of the paper is Theorem 1 which states that in this relative situation an Atiyah-Bott-Lefschetz-like fixed point formula can be obtained such that it contains two summations over the fixed points of \(g\): one term is a usual summation over the fixed points of \(g\) over \(M\) and the other term is a usual summation over the fixed points of \(g\) over \(X\) (see Theorem 1 for the precise formulation). The theorem is then applied to obtain trace formulas for relative cone complexes and for the relative de Rham complex (see Section 3 in the paper).