Connected sum decompositions of high-dimensional manifolds

Abstract

The classical Kneser-Milnor theorem says that every closed oriented connected 3-dimensional manifold admits a unique connected sum decomposition into manifolds that cannot be decomposed any further. We discuss to what degree such decompositions exist in higher dimensions and we show that in many settings uniqueness fails in higher dimensions.

arXiv comment25 pages, fixed several minor mistakes, final version

References in corpus (8)

• A survey of the foundations of four-manifold theory in the topological category
• Realizing algebraic 2-complexes by cell complexes
• Detecting exotic spheres in low dimensions using coker J
• An exotic presentation of Q_28
• A stably free nonfree module and its relevance for homotopy classification, case Q_{28}
• A cancellation theorem for modules over integral group rings
• Cancellation for $(G,n)$-complexes and the Swan finiteness obstruction
• Inertia groups and smooth structures of $(n-1)$-connected $2n$-manifolds

Cited by in corpus (7)

• An exotic presentation of Q_28
• Projective modules and the homotopy classification of $(G,n)$-complexes
• Cancellation for $(G,n)$-complexes and the Swan finiteness obstruction
• Asymptotic mapping class groups of Cantor manifolds and their finiteness properties
• The decomposition of the higher-order homology embedding constructed from the $k$-Laplacian
• Manifolds with small topological complexity
• The classification of surfaces via normal curves