Suspension Splitting and Cohomotopy Sets of Simply Connected 7-manifolds

Let be a closed simply connected 7-manifold.In this paper we establish homotopy decompositions of the reduced suspension space into a wedge sum of simpler spaces when localized at a set of primes.These decompositions are applied to study the cohomotopy sets ( ) and the -local cohomotopy sets 4 ( ; Z ( ) ). IntroductionSeven-dimensional manifolds play a central role in geometry and topology.In differential geometry, for example, there has been growing interest in 2 -manifolds, particularly following the seminal work of Donaldson and Thomas [11].Changing the focus to the algebraic topology of gauge theory, a general principle [37,34,17] posits that the homotopy theory of gauge groups is closely related to the homotopy type of the underlying manifolds.In geometric topology, the foundational contributions by Kreck [23] and Crowley-Nordstrm [9] independently established a complete classification of closed smooth 2-connected 7-manifolds using systems of algebraic invariants.Building on his earlier breakthrough work [22], Kreck [23] further extended this classification to simply connected 7-manifolds with torsion-free second homology.In recent years, the homotopy theory of suspended manifolds has attracted significant interest among topologists.For example, So and Theriault [35] and Li [24] determined the homotopy type of suspended 4-manifolds.Subsequent work by Huang [16], Li and Zhu [26], and Amelotte, Cutler, and So [1] extended this analysis to certain 5-manifolds.Huang [18] and Cutler and So [10] further advanced these results for 6manifolds.Additionally, Membrillo-Solis [28] investigated the problem in the context of 3 -bundles over 4 at large primes.Collectively, these studies have applications in characterizing key invariants in geometry and mathematical physics, such as reduced -groups and gauge groups.This paper focuses on the homotopy type of the (reduced) suspension space of 7manifolds.Let be a closed simply connected 7-manifold whose reduced integral homology groups * (; Z) are given by the following table: 2 3 4 5 0, 7 otherwise (; Z) Z Z Z Z Z 0 , (1.1)