Related papers: Real Bott manifold structure of $n$-dimensional Kl…
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…
In this short note, we compute the Betti numbers of the moduli stack of flat SU(3)-bundles over a Klein bottle. We also handle the general compact group case over RP^2. In all cases the cohomology is found to be equivariantly formal,…
A small cover is a closed smooth manifold of dimension $n$ having a locally standard $\mathbb{Z}_2^n$-action whose orbit space is isomorphic to a simple polytope. A typical example of small covers is a real projective toric manifold (or,…
Bredon and Wood have given a complete answer to the embeddability question for nonorientable surfaces in lens spaces. They formulate their result in terms of a recursive formula that determines, for a given lens space, the minimal genus of…
We point out a gap in the proof of the Davis--Januszkiewicz theorem on cohomology of small covers of simple polytopes, and give a correction to this proof. We use this theorem to compute explicitly the Betti numbers for a wide class of…
In this note, a general result for determining the rational hulls of fibered sets in $\mathbb{C}^2$ is established. We use this to compute the rational hull of Rudin's Klein bottle, the first explicit example of a totally real nonorientable…
A real Bott tower is obtained as the orbit space of the $n$-torus $T^n$ by the free action of an elementary abelian 2-group $(\mathbb{Z}_2)^n$. This paper deals with the classification of 5-dimensional real Bott towers and study certain…
Together with the Moebius strip, the Klein bottle is one of the intriguing objects in the universe of geometry, sometimes appearing in non-mathematical contexts too. Until now, several parametrizations of it as a surface immersed in…
We explore a higher-dimensional universe that is a product of Minkowski space and the nonorientable Klein bottle. The topology explicitly breaks important symmetries, such as translational invariance and (5+1)-dimensional CP invariance.…
We define a notion of facets-pairing structure and its seal space on a nice manifold with corners. We will study facets-pairing structures on any cube in detail and investigate when the seal space of a facets-pairing structure on a cube is…
We show that all the small covers which are infra-nilmanifolds are exactly real Bott manifolds. This implies that any small cover which admits a flat Riemannian metric must be a real Bott manifold. In addition, we will study small covers…
We show that the number of marked minimal models of an n-dimensional smooth complex projective variety of general type can be bounded in terms of its volume, and, if n=3, also in terms of its Betti numbers. For an n-dimensional projective…
Moduli spaces of real bundles over a real curve arise naturally as Lagrangian submanifolds of the moduli space of semi-stable bundles over a complex curve. In this paper, we adapt the methods of Atiyah-Bott's "Yang-Mills over a Riemann…
We construct modular spaces of all 6-dimensional real semisimple Drinfeld doubles, i.e. the sets of all possible decompositions of the Lie algebra of the Drinfeld double into Manin triples. These modular spaces are significantly different…
The geodesic complexity of a metric space X is the smallest k for which there is a partition of X x X into ENRs E_0,...,E_k on each of which there is a continuous choice of minimal geodesic sigma(x_0,x_1) from x_0 to x_1. We prove that the…
We present numerical polyhedron data for the image of a piecewise-linear map from a zero-curvature Klein bottle into Euclidean 3-space such that every point in the domain has a neighborhood which is isometrically embedded. To the author's…
We estimate from below the expected Betti numbers of real hypersurfaces taken at random in a smooth real projective n-dimensional manifold. These random hypersurfaces are chosen in the linear system of a large d-th power of a real ample…
A quasitoric manifold (resp. a small cover) is a $2n$-dimensional (resp. an $n$-dimensional) smooth closed manifold with an effective locally standard action of $(S^1)^n$ (resp. $(\mathbb Z_2)^n$) whose orbit space is combinatorially an…
In this paper, we compute the LS-category and equivariant LS-category of a small cover and its real moment angle manifold. We calculate a tight lower bound for the topological complexity of many small covers over a product of simplices.…
We propose a high dimensional generalisation of the standard Klein bottle, going beyond those considered previously. We address the problem of generating continuous scalar fields (distributions) and dynamical systems (flows) on such state…