The motivation for this work was a calculation by Oshanin of the image of the signature homomorphism from the special unitary cobordism ring into the integers. Here we compute this image for symplectic cobordism. This is accomplished by proving two divisibility theorems and then giving examples to show the theorems are the best possible.

Let R be a commutative ring. The cup-length of R is defined by the greatest number n such that there exist x1, . . . , xn ∈ R \ R with x1 · · · xn , 0. We denote the cup-length of R by cup(R). In particular, for a space X and a commutative ring A, the cup-length of X with the coefficient A, is defined by cup(H̃(X; A)). We denote it by cupA(X). It is well-known that cupA(X) is a lower bound for t...

Hom (G,H) is a polyhedral complex defined for any two undirected graphs G and H. This construction was introduced by Lovász to give lower bounds for chromatic numbers of graphs. In this paper we initiate the study of the topological properties of this class of complexes. We prove that Hom (Km,Kn) is homotopy equivalent to a wedge of (n−m)dimensional spheres, and provide an enumeration formula f...

Applications of multiple input and multiple output (MIMO) with Stiefel manifold are growing in the next generation communications and their system engineering developments. These applications need to be optimized efficiently with less complexity and cost. In this research, optimization problems in MIMO systems using Stiefel manifold are considered. As far as general manifolds are concerned, opt...

Let ( X , J ) be an almost complex manifold with a (smooth) involution σ : → such that Fix ≠ ∅ . Assume is conjugation, i.e., the differential of anti-commutes The space P m = S × / ∼ where v x − was referred to as generalized Dold manifold. above definition admits obvious generalization much wider class spaces are arbitrary topological spaces. resulting will called space. When and CW complexes...

A homomorphism J: 7Tk_1(SOw) -> nm+k_1(S ) from the homotopy groups of rotation groups to the homotopy groups of spheres has been defined by H. Hopf and G. W. Whitehead. This homomorphism plays an important role in the study of differentiable manifolds. We will study its relation to one particular problem: the question of possible Pontrjagin numbers of an 'almost parallelizable' manifold. Defin...

The study of higher-order and real topological states as well the material realization have become a research forefront condensed matter physics in recent years. Twisted bilayer graphene (tbG) is proved to topology. However, whether this conclusion can be extended other two-dimensional twisted carbon materials mechanism behind it lack explorations. In paper, we identify $\ensuremath{\alpha}$-gr...

Optimization on Stiefel manifolds was discussed by Rapcsák in earlier papers, and some global optimization methods were considered and tested on Stiefel manifolds. In the paper, test functions are given with known global optimum points and their optimal function values. A restriction, which leads to a discretization of the problem is suggested, which results in a problem equivalent to the well-...

In their paper on discrete analogues of some classical systems such as the rigid body and the geodesic flow on an ellipsoid, Moser and Veselov introduced their analysis in the general context of flows on Stiefel manifolds. We consider here a general class of continuous time, quadratic cost, optimal control problems on Stiefel manifolds, which in the extreme dimensions again yield these classica...

With methods from Riemannian geometry bounds for the minimal distance of packings in the Graßmann and Stiefel manifolds will be derived and analysed, revealing implications for certain aspects of coding theory. Though motivated from coding theory on the Graßmann and Stiefel manifolds, the method can be generalised to arbitrary Riemannian homogeneous spaces as well.