Rational cobordisms and integral homology

We consider the question of when a rational homology $3$-sphere is rational homology cobordant to a connected sum of lens spaces. We prove that every rational homology cobordism class in the subgroup generated by lens spaces is represented by a unique connected sum of lens spaces whose first homology group injects in the first homology group of any other element in the same class. As a first consequence, we show that several natural maps to the rational homology cobordism group have infinite-rank cokernels. Further consequences include a divisibility condition between the determinants of a connected sum of $2$-bridge knots and any other knot in the same concordance class. Lastly, we use knot Floer homology combined with our main result to obstruct Dehn surgeries on knots from being rationally cobordant to lens spaces.

The fundamental class of smooth Kuranishi atlases with trivial isotropy

Kuranishi structures were introduced in the 1990s by Fukaya and Ono for the purpose of assigning a virtual cycle to moduli spaces of pseudoholomorphic curves that cannot be regularized by geometric methods. Their core idea was to build such a cycle by patching local finite dimensional reductions. The first sections of this paper discuss topological, algebraic and analytic challenges that arise in this program.We then develop a theory of Kuranishi atlases and cobordisms that transparently resolves these challenges, for simplicity concentrating on the case of trivial isotropy. In this case, we assign to a cobordism class of additive weak Kuranishi atlases both a virtual moduli cycle (VMC — a cobordism class of smooth manifolds) and a virtual fundamental class (VFC — a Cech homology class). We, moreover, show that such Kuranishi atlases exist on simple Gromov–Witten moduli spaces and develop the technical results in a manner that easily transfers to more general settings.

BOUNDS ON THE DIMENSION OF MANIFOLDS WITH INVOLUTION FIXING Fn ∪ F2

Let Mm be a closed smooth manifold with an involution having fixed point set of the form Fn ∪ F2, where Fn and F2 are submanifolds with dimensions n and 2, respectively, where n ≥ 4 is even (n < m). Suppose that the normal bundle of F2 in Mm, μ → F2, does not bound, and denote by β the stable cobordism class of μ → F2. In this paper, we determine the upper bound for m in terms of the pair (n, β) for many such pairs. The similar question for n odd (n ≥ 3) was completely solved in a previous paper of the authors. The existence of these upper bounds is guaranteed by the famous 5/2-theorem of Boardman, which establishes that, under the above hypotheses, m ≤ 5/2n.

Topology of four-manifolds with special homotopy groups

We study the homotopy type and the s—cobordism class of a closed connected topological 4-manifold with vanishing second homotopy group. Our results are related to problem 4.53 of Kirby in Geometric Topology, Studies in Advanced Math. 2 (1997), and give a partial answer to a question stated by Hillman in Bull. London Math. Soc.27 (1995) 387–391.

Cutting and Pasting Zp-Manifolds(1)

Let Mn and Nn be n-dimensional closed smooth oriented Zp-manifolds where p is an odd prime and Zp is the cyclic group of order p. This paper determines necessary and sufficient conditions under which Mn and Nn are equivalent under a special equivariant cut and past equivalence.The only invariants are (a) the Euler characteristics of the Zp-manifolds, (b) the Euler characteristics of the fixed point manifolds in each fixed point dimesnion with specified normal representations, and (c) the oriented Zp-stratified cobordism class of the Zp-manifolds.