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A central problem in low-dimensional topology asks which homology $3$-spheres bound contractible $4$-manifolds or homology $4$-balls. In this paper, we address this question for plumbed $3$-manifolds and we present two new infinite…
Twist tori are examples of exotic monotone lagrangian tori, presented in [1]. This tree of examples grew up over the first one --- the torus $\Theta \in \R^4$, constructured in [2] and [3]. On the other hand, in [4] and [5] we proposed a…
We apply our earlier work on the higher-dimensional analogue of the Mumford conjecture to two questions. Inspired by work of Ebert we prove non-triviality of certain characteristic classes of bundles of smooth closed manifolds. Inspired by…
We prove the Lipman-Zariski conjecture for complex surface singularities of genus one, and also for those of genus two whose link is not a rational homology sphere. As an application, we characterize complex $2$-tori as the only normal…
By using two different invariants for the Rubik's Magic puzzle, one of metric type, the other of topological type, we can dramatically reduce the universe of constructible configurations of the puzzle. Finding the set of actually…
This small note presents in any dimension a family of cubics that have finite-dimensional motive (in the sense of Kimura). As an illustration, we verify a conjecture of Voevodsky for these cubics, and a conjecture of Murre for the Fano…
For a given $w$ in a Coxeter group $W$ the elements $u$ smaller than $w$ in Bruhat order can be seen as the end-alcoves of stammering galleries of type $w$ in the Coxeter complex $\Sigma$. We generalize this notion and consider sets of…
For a 3-dimensional manifold $M^3$, its complexity $c(M^3)$, introduced by S.Matveev, is the minimal number of vertices of an almost simple spine of $M^3$; in many cases it is equal to the minimal number of tetrahedra in a singular…
A fundamental result in 4-manifold topology asserts that any two exotic smooth structures on a simply-connected, closed 4-manifold differ by a cork twist: the operation of removing a compact, contractible, codimension-zero submanifold and…
We introduce a new generalization of Gompf nuclei and give applications. We construct infinitely many exotic smooth structures for a large class of compact 4-manifolds with boundary, regarding topological invariants. We prove that a large…
The construction of new hyper-Kaehler manifolds by taking the infinite monopole mass limit of certain Bogomol'nyi-Prasad-Sommerfield monopole moduli spaces is considered. The one-parameter family of hyperkaehler manifolds due to Dancer is…
In the literature, there are two different notions of pseudosymmetric manifolds, one by Chaki [7] and other by Deszcz [16], and there are many papers related to these notions. The object of the present paper is to deduce necessary and…
We construct a new twenty-dimensional family of projective eight-dimensional irreducible holomorphic symplectic manifolds: the compactified moduli space M_3(Y) of twisted cubics on a smooth cubic fourfold Y that does not contain a plane is…
The first example of a compact manifold admitting both complex and symplectic structures but not admitting a K\"ahler structure is the renowned Kodaira-Thurston manifold. We review its construction and show that this paradigm is very…
Turaev Viro invariants are amongst the most powerful tools to distinguish 3-manifolds: They are implemented in mathematical software, and allow practical computations. The invariants can be computed purely combinatorially by enumerating…
Turaev's shadow can be seen locally as the Stein factorization of a stable map. In this paper, we define the notion of stable map complexity for a compact orientable 3-manifold bounded by (possibly empty) tori counting, with some weights,…
Let $E$ be an elliptic curve defined over $\mathbb{Q}$ with supersingular reduction at $p \geq 5$, and $K$ be an imaginary quadratic field such that $p$ is inert in $K/\mathbb{Q}$. In this paper, we prove the analogous of the ``weak''…
A large number of examples of compact $G_2$ manifolds, relevant to supersymmetric compactifications of M-Theory to four dimensions, can be constructed by forming a twisted connected sum of two appropriate building blocks times a circle.…
We prove that any closed simply-connected smooth 4-manifold is 16-fold branched covered by a product of an orientable surface with the 2-torus, where the construction is natural with respect to spin structures. In particular this solves…
HyperK\"ahler spaces, including manifolds, orbifolds and conical singularities play an important role in superstring/$M$-theory and gauge theories as well as in differential and algebraic geometry. In this paper we provide hundreds of new…