Rational homotopy type of the embedding maps between total spaces of the projectivization bundles over complex projective spaces

Source abstract

Purpose This paper investigates the embedding maps fn,k:P(En) ↪ P(En+k), where P(En) and P(En+k) are the total spaces of the projectivized tangent bundles over CPn and CPn+k, respectively. We determine the rational homotopy type of the embedding complement C(fn,k). Design/methodology/approach Using Sullivan models, relative Sullivan algebras, Poincaré duality and cohomological shriek maps, we construct a commutative differential graded algebra (CDGA) model for the complement of the embedding map fn,k. The algebraic mapping cone associated with the shriek map is used to identify the rational homotopy type of the complement. Findings The embedding complement C(fn,k) is formal and admits the CDGA model (Λ(a2,b2)/(∑i=0n+ka2ib2 n+k−i,a2kb2k,b2 n+k+1),0). Moreover, there exists a rational map CPk−1×CPk−1→C(fn,k) which induces a surjective map in cohomology. Originality/value The paper provides an explicit algebraic model for the complements of embeddings between total spaces of projectivized tangent bundles and establishes a connection between these complements, Poincaré embedding theory and products of complex projective spaces. The resulting models provide an effective algebraic framework for studying the rational cohomology and rational homotopy properties of these embedding complements.