In this paper we consider a Procrustes problem on the set of sub-Stiefel matrices. A sub-Stiefel matrix is a matrix that results from deleting simultaneously the last row and the last column of an orthogonal matrix. An iterative algorithm for computing the solution of the sub-Stiefel Procrustes problem is proposed. For these purposes we investigate the properties of sub-Stiefel matrices. We als...

In this paper, nonholonomic gerbes will be naturally derived for manifolds and vector bundle spaces provided with nonintegrable distributions (in brief, nonholonomic spaces). An important example of such gerbes is related to distributions defining nonlinear connection (N–connection) structures. They geometrically unify and develop the concepts of Riemann–Cartan manifolds and Lagrange–Finsler sp...

To any two graphs G and H one can associate a cell complex Hom (G,H) by taking all graph multihomorphisms from G to H as cells. In this paper we prove the Lovász Conjecture which states that if Hom (C2r+1, G) is k-connected, then χ(G) ≥ k + 4, where r, k ∈ Z, r ≥ 1, k ≥ −1, and C2r+1 denotes the cycle with 2r + 1 vertices. The proof requires analysis of the complexes Hom (C2r+1,Kn). For even n,...

These are lecture notes for a series of five lectures I gave to other graduate students about characteristic classes through UT Austin’s summer minicourse program (see https://www.ma.utexas.edu/users/ richard.wong/Minicourses.html for more details). Beware of potential typos. In these notes I cover the basic theory of Stiefel-Whitney, Wu, Chern, Pontrjagin, and Euler classes, introducing some i...

We expose the theory of the construction of a natural G 2 structure on the unit sphere tangent bundle SM of any given orientable Riemannian 4-manifold M , as laid in [4]. This time we work in the context of metric connections, or geometry with torsion. 1 Recalling the theory By a G 2 manifold it is understood a 7 dimensional Riemannian manifold with holonomy group contained in G 2 = Aut(O). The...

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...

To any two graphs G and H one can associate a cell complex Hom (G,H) by taking all graph multihomomorphisms from G to H as cells. In this paper we prove the Lovász conjecture which states that if Hom (C2r+1, G) is k-connected, then χ(G) ≥ k + 4, where r, k ∈ Z, r ≥ 1, k ≥ −1, and C2r+1 denotes the cycle with 2r+1 vertices. The proof requires analysis of the complexes Hom (C2r+1,Kn). For even n,...

LetGk,n be the Grassmannmanifold of k-planes inR . Borel showed that H∗ (Gk,n; Z2) = Z2 [w1, . . . , wk] /Ik,n where Ik,n is the ideal generated by the dual Stiefel-Whitney classes wn+1, . . . , wn+k. We compute Groebner bases for the ideals I2,2i−3 and I2,2i−4 and use these results along with the theory of modi ed Postnikov towers to prove new immersion results, namely that G2,2i−3 immerses in...