Certain generalized higher derived functors associated to quasitoric manifolds (Q721979)

In this paper right derived functors of the primitive element functor defined on \(R\)-coalgebras are studied. Here \(R\) is a graded commutative ring with unit. The results are applied to the study of quasitoric manifolds. The main results are as follows: Theorem. Let \(R\) be a ring. Suppose \(C_1\) and \(C_2\) are \(\mathbb{Z}\)-coalgebras. If \(R^iP(C_1;\mathbb{Z})\cong R^iP(C_2;\mathbb{Z})\) for \(i>k\), then \(R^iP(C_1\otimes R;R)\cong R^iP(C_2\otimes R;R)\) for \(i>k\). Here \(R^iP(C;R)\) denotes the \(i\)-th right derived functor of the primitive element functor of the \(R\)-coalgebra \(C\). Theorem. Let \(E\) be a complex orientable theory with coefficients concentrated in even degrees. Suppose \(M_1\) and \(M_2\) are two quasitoric manifolds with orbit space a simple convex polytope \(P\), then \(R^iP(E_*(M_1);E_*)\cong R^iP(E_*(M_2);E_*)\) for \(i>1\). Applications of the latter result to rigidity problems in toric topology are also discussed.