Mathematical Sciences: Decompositions in Dimension Four, Local Connectivity at Decomposition Spaces, Homogeneity and Generalized Manifolds

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Professor Walsh will continue to investigate cell-like decompositions in dimension four with the ultimate goal of providing a characterization of four-manifolds comparable to that available in higher dimensions. Current focus is on a version of a disjoint-disks-property that, in spite of its apparent simplicity, can be extremely difficult to detect even in relatively "simple" examples. The project includes investigating the cohomological local connectedness of decomposition spaces associated to decompositions consisting of absolute neighborhood retracts and investigating the local homology of compact finite- dimensional absolute neighborhood retracts with the goal of resolving the conjecture of Bing-Borsuk that homogeneous finite- dimensional absolute neighborhood retracts are generalized manifolds.