Borel Conjecture and Dual Borel Conjecture

We show that it is consistent that the Borel Conjecture and the dual Borel Conjecture hold simultaneously.Comment: 47 pages, revised version 2013 (some typos removed, some points elaborated. Dedication added.

Invariant subspace problem in Hilbert space: Correlation with the Kadison-Singer problem and the Borel conjecture

This paper explores the intriguing connections between the invariant subspace problem, the Kadison-Singer problem, and the Borel conjecture. The Kadison-Singer problem, originally formulated in terms of pure states on C*-algebras, was later reformulated using projections, establishing a link with the invariant subspace problem. The Borel conjecture, a question in descriptive set theory, connects to the invariant subspace problem through Borel equivalence relations. This paper elucidates these connections, underscoring the interplay of unsolved mathematical problems and the collaborative nature of mathematical research

Homotopy invariance of 4-manifold decompositions: connected sums

We show, up to h-cobordism, that the existence and uniqueness of connected sum decompositions of oriented 4-dimensional manifolds is an invariant of homotopy equivalence, assuming that the fundamental group of each summand is "good" in the sense of Freedman and Quinn. On a separate note, we observe that the Borel Conjecture is true in dimension 4 up to s-cobordism, assuming that the fundamental group satisfies the Farrell--Jones Conjecture.Comment: 14 pages, 1 figure, accepted by Topology and its Application