Related papers: Indecomposable Klein bottles with order-4 meridian…

It is shown that an embedded Lagrangian Klein bottle represents a non-trivial mod 2 homology class in a compact symplectic four-manifold $(X,\omega)$ with $c_1(X)\cdot[\omega]>0$. (In versions 1 and 2, the last assumption was missing. A…

Let $K$ be a Klein bottle. We show that the infimum of the Willmore energy among all immersed Klein bottles in Euclidean $n$-space is attained by a smooth embedded Klein bottle, where $n\geq 4$. There are three distinct regular homotopy…

Using a recent result of Bowden, Hensel and Webb, we prove the existence of a homeomorphism with positive stable commutator length in the group of homeomorphisms of the Klein bottle which are isotopic to the identity.

We construct a family of smooth branched coverings of degree $2$ of the sphere $S^2$ having a completely invariant indecomposable continuum $K$ and infinitely many Wada Lakes.

Steiner systems are a fascinating topic of combinatorics. The most studied Steiner systems are $S(2, 3, v)$ (Steiner triple systems), $S(3, 4, v)$ (Steiner quadruple systems), and $S(2, 4, v)$. There are a few infinite families of Steiner…

We show that there are 4 infinite families of lattice equable kites, given by corresponding Pell or Pell-like equations, but up to Euclidean motions, there are exactly 5 lattice equable trapezoids (2 isosceles, 2 right, 1 singular) and 4…

In this paper, we construct almost resolvable cycle systems of order $4k+1$ for odd $k\ge 11$. This completes the proof of the existence of almost resolvable cycle systems with odd cycle length.

We describe the singularities and related properties of the arrangement of 21 reducible polars of Klein's quartic, containing Klein's well-known arrangement of $21$ lines.

In this paper, we give an example of a chiral 4-polytope in projective 3-space. This example naturally yields a finite chiral 4-polytope in Euclidean 4-space, giving a counterexample to Theorem 11.2 of [2].

An n-dimensional analogue of the Klein bottle arose in our study of topological complexity of planar polygon spaces. We determine its integral cohomology algebra and stable homotopy type, and give an explicit immersion and embedding in…

We study the structure of the codifferent and of additively indecomposable integers in families of totally real cubic fields. We prove that for cubic orders in these fields, the minimal trace of indecomposable integers multiplied by totally…

Suppose that $X,X'$ are simply-connected closed exotic 4-manifolds. It is well-known that $X'$ is obtained by an order 2 cork twist of $X$. We give an infinite exotic family of 4-manifolds not generated by any infinite order cork. This is…

Suppose you have a nonorientable Lagrangian surface L in a symplectic 4-manifold. How far can you deform the symplectic form before the smooth isotopy class of L contains no Lagrangians? I solve this question for a particular Lagrangian…

It is shown that the n-dimensional Klein bottle admits a Lagrangian embedding into R^{2n} if and only if n is odd.

Examples are given to show that some compact contractible 4-manifolds can be knotted in the 4-sphere. It is then proved that any finitely presented perfect group with a balanced presentation is a knot group for an embedding of some…

We construct a nonseparable Banach space $\mathcal X$ (actually, of density continuum) such that any uncountable subset $\mathcal Y$ of the unit sphere of $\mathcal X$ contains uncountably many points distant by less than $1$ (in fact, by…

Given a coloring of the k-element subsets of an uncountable separable metric space, we show that there exists an infinite monochromatic subset which contains its limit point.

We prove that if the order of the first homology of the 2-fold branched cover of a knot K in the 3-sphere is given by pm where p is a prime congruent to 3 mod 4 and gcd(p,m) =1, then K is of infinite order in the knot concordance group.…

We construct an infinitesimal invariant for cycles in a family with cohomology class in the total space lying in a given level of the Leray filtration. This infinitesimal invariant detects cycles modulo algebraic equivalence in the fibers.…

Let K be a set of infinite cardinals such that the cardinality of K is the first strong limit cardinal greater than uncountably many strong limit cardinals. We construct a family of pairwise non-embeddable groups which contains 2^k groups…