In this paper we prove the Lovász Conjecture: If Hom (C2r+1,H) is k-connected, then χ(H) ≥ k + 4, where H is a finite undirected graph, C2r+1 is a cycle with 2r+1 vertices, r, k ∈ Z, r ≥ 1, k ≥ −1, and Hom (G,H) is the cell complex with the vertex set being the set of all graph homomorphisms from G to H , and cells all allowed list H-colorings of G. Our method is to compute, by means of spectra...

By the early 1950’s algebraic topology had reached great heights with Serre’s thesis and the calculations in the seminar of Henri Cartan of the cohomology of spaces with one nonzero homotopy group in terms of Steenrod operations. There was also the appearance of the new characteristic classes of vector bundles, Pontryagin classes (Z coefficients) and Chern-Weil classes (coefficients in R) joini...

This paper is on the homotopy classification of maps of (n+1)-dimensional manifolds into the n-dimensional sphere. For a continuous map f : M → S define the degree deg f ∈ H1(M;Z) to be the class dual to f[S], where [S] ∈ H(S;Z) is the fundamental class. We present a short and direct proof of the following specific case of the Pontryagin-Steenrod-Wu theorem: Theorem. Let M be a connected orient...

0 Introduction The purpose of this preprint is to construct a new invariant of the smooth structure of a simply connected 4-manifold M so called Spin-polynomials γ g,C and to show how to use it to compare the smooth structures of rational surfaces and surfaces of general type. This Spin-polynomial (0.1) is the analogue of the original Donaldson polynomial γ g (2, c 1 , c 2) ∈ S d H 2 (M, Z Z) (...

of the almost-complex manifold (X, J). The only obstruction [10] to the existence of an almost-complex structure J on X is that X be spinc. This happens precisely when the second Stiefel-Whitney class w2(X) ∈ H(X,Z2) can be written as the mod-2 reduction of an element of H2(X,Z), in which case each preimage of w2 in H2(X,Z) can be realized as c1 for some almostcomplex structure J . It follows t...

We present a novel class of topological insulators, termed the Takagi insulators (TTIs), which is protected by sublattice symmetry and spacetime inversion ($\mathcal P\mathcal T$) symmetry. The required symmetries for TTIs can be realized on any bipartite lattice where exchanges sublattices. protecting lead to classifying space Hamiltonians being unitary symmetric matrices, therefore Takagi's f...

For a representation of a matroid the combinatorially defined Kazhdan-Lusztig polynomial computes the intersection cohomology of the associated reciprocal plane. However, these polynomials are difficult to compute and there are numerous open conjectures about their structure. For example, it is unknown whether or not the coefficients are non-negative for non-representable matroids. The main res...

A low complexity Lie group method for numerical integration of ordinary differential equations on the orthogonal Stiefel manifold is presented. Based on the quotient space representation of the Stiefel manifold we provide a representation of the tangent space suitable for Lie group methods. According to this representation a special type of generalized polar coordinates (GPC) is defined and use...

Optimization on Stiefel manifolds was discussed by Rapcsák in earlier papers. There, some numerical methods of global optimization are dealt with and tested on Stiefel manifolds. In the paper the structure of the optimizer points is given in some particular problem instances and for a special form of a quadratic problem defined on a Stiefel manifold. Some reduction tricks and results are obtain...

In this note, we characterize the convex hull of the Stiefel manifold and we find its strong convexity parameter. We also introduce the notion of roundness of a set and show that the Stiefel manifold is round.