Nonsplittability of the rational homology cobordism group of 3-manifolds
The three dimensional homology cobordism group is very large, and only partially understood. This paper proves new results about the smooth rational homology cobordism group. In particular one may consider the relation between the smooth cobordism groups with \(\mathbb{Z}[1/p]\) coefficients and the rational homology cobordism group. It would be reasonable to conjecture that the topological rational homology cobordism group is the direct sum of the topological \(\mathbb{Z}[1/p]\) homology cobordism groups classified by the Witt groups. This is far from the case for the smooth homology cobordism groups. Indeed earlier work shows that the kernel of the map taking the sum of \(p\)-cobordism groups into the rational homology cobordism group is infinitely generated. This paper shows that the cokernel of this map contains an infinite rank free group as well as in infinite subgroup generated by elements of order two. The kernel contains an infinite subgroup generated by \(3\)-mainfolds that topologically bound rational balls. The results are obtained via Heegaard-Floer theory.
scientific article; zbMATH DE number 6369633