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Register a new userConnected sum decompositions of high-dimensional manifolds
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The classical Kneser-Milnor theorem says that every closed oriented connected 3-dimensional manifold admits a unique connected sum decomposition into manifolds that cannot be decomposed any further. We discuss to what degree such decompositions exist in higher dimensions and we show that in many settings uniqueness fails in higher dimensions.
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We present a classification theorem for closed smooth spin 2-connected 7-manifolds M. This builds on the almost-smooth classification from the first authors thesis. The main additional ingredient is an extension of the Eells-Kuiper invariant for any closed spin 7-manifold, regardless of whether the spin characteristic class p_M in the fourth integral cohomology of M is torsion. In addition we determine the inertia group of 2-connected M - equivalently the number of oriented smooth structures on the underlying topological manifold - in terms of p_M and the torsion linking form.
We study the set $widehat{mathcal S}_M$ of framed smoothly slice links which lie on the boundary of the complement of a 1-handlebody in a closed, simply connected, smooth 4-manifold $M$. We show that $widehat{mathcal S}_M$ is well-defined and describe how it relates to exotic phenomena in dimension four. In particular, in the case when $X$ is smooth, with a handle decompositions with no 1-handles and homeomorphic to but not smoothly embeddable in $D^4$, our results tell us that $X$ is exotic if and only if there is a link $Lhookrightarrow S^3$ which is smoothly slice in $X$, but not in $D^4$. Furthermore, we extend the notion of high genus 2-handle attachment, introduced by Hayden and Piccirillo, to prove that exotic 4-disks that are smoothly embeddable in $D^4$, and therefore possible counterexamples to the smooth 4-dimensional Schonflies conjecture, cannot be distinguished from $D^4$ only by comparing the slice genus functions of links.
For every $k geq 2$ and $n geq 2$ we construct $n$ pairwise homotopically inequivalent simply-connected, closed $4k$-dimensional manifolds, all of which are stably diffeomorphic to one another. Each of these manifolds has hyperbolic intersection form and is stably parallelisable. In dimension $4$, we exhibit an analogous phenomenon for spin$^{c}$ structures on $S^2 times S^2$. For $mgeq 1$, we also provide similar $(4m{-}1)$-connected $8m$-dimensional examples, where the number of homotopy types in a stable diffeomorphism class is related to the order of the image of the stable $J$-homomorphism $pi_{4m-1}(SO) to pi^s_{4m-1}$.
We prove a connected sum formula for involutive Heegaard Floer homology, and use it to study the involutive correction terms of connected sums. In particular, we give an example of a three-manifold with $underline{d}(Y) eq d(Y) eq overline{d}(Y)$. We also construct a homomorphism from the three-dimensional homology cobordism group to an algebraically defined Abelian group, consisting of certain complexes (equipped with a homotopy involution) modulo a notion of local equivalence.
In this paper we study smooth orientation-preserving free actions of the cyclic group $mathbb Z/m$ on a class of $(n-1)$-connected $2n$-manifolds, $sharp g (S^n times S^n)sharp Sigma$, where $Sigma$ is a homotopy $2n$-sphere. When $n=2$ we obtain a classification up to topological conjugation. When $n=3$ we obtain a classification up to smooth conjugation. When $n ge 4$ we obtain a classification up to smooth conjugation when the prime factors of $m$ are larger than a constant $C(n)$.
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