We consider closed topological 4-manifolds M with universal cover S × S and Euler characteristic χ(M) = 1. All such manifolds with π = π1(M) ∼= Z/4 are homotopy equivalent. In this case, we show that there are four homeomorphism types, and propose a candidate for a smooth example which is not homeomorphic to the geometric quotient. If π ∼= Z/2×Z/2, we show that there are three homotopy types (a...

Let Y be a closed oriented three-manifold and K be a framed knot in Y . For a rational number r , let Yr(K) be the resulting manifold obtained by Dehn surgery along K with slope r with respect to the given framing. For a knot K in S3 , we will always use the Seifert framing. Two surgeries along K with distinct slopes r and s are called cosmetic if Yr(K) and Ys(K) are homeomorphic. The two surge...

We say that an arbitrary manifold (M,∂M) is topologically rigid relative to its ends if it satisfies the following condition. If (N, ∂N) is any other manifold with a compact subset C ⊂ N for which a proper homotopy equivalence h : (N, ∂N)→ (M,∂M) is a homeomorphism on ∂N ∪ (N\C), then there is a compact subset K ⊂ N and a proper homotopy ht : (N, ∂N) → (M,∂M) from h to a homeomorphism such that...

This paper is devoted to characterizing so-called order isomorphisms intertwining the $$L^2$$ -semigroups of two quasi-regular Dirichlet forms. We first show that every unitary isomorphism semigroups composition h-transformation and quasi-homeomorphism. In addition, under assumption underlying spaces admit irreducible decompositions for forms, (not necessarily unitary) can be expressed as h-tra...