arXiv subjects
Publications and source records attributed to Vladimir Grujic.
We introduce a diagrammatic language for compact, orientable 3-dimensional manifolds with boundary. A diagrammatic calculus (both integral and rational version) appropriate for this language is introduced and its completeness is proved in the paper. Moreover, a calculus consisting of a finite list of local moves is presented.
We construct small covers and quasitoric manifolds over $n$-dimensional simple polytopes which allow proper colorings of facets with $n$ colors. We calculate Stiefel-Whitney classes of these manifolds as obstructions to immersions and embeddings into Euclidean spaces. The largest dimension required for embedding is achieved in the case $n$ is a power of two.
The generalized Dehn-Sommerville relations determine the odd subalgebra of the combinatorial Hopf algebra. We introduce a class of eulerian hypergraphs that satisfy the generalized Dehn-Sommerville relations for the combinatorial Hopf algebra of hypergraphs. We characterize a wide class of eulerian hypergraphs according to the combinatorics of underlying clutters. The analogous results hold for simplicial complexes by the isomorphism which is induced from the correspondence of clutters and simplicial complexes.
For a finite simplicial complex K and a CW-pair (X,A), there is an associated CW-complex Z_K(X,A), known as a polyhedral product. We apply discrete Morse theory to a particular CW-structure on the n-sphere moment-angle complexes Z_K(D^{n}, S^{n-1}). For the class of simplicial complexes with vertex-decomposable duals, we show that the associated n-sphere moment-angle complexes have the homotopy type of wedges of spheres. As a corollary we show that a sufficiently high suspension of any restriction of a simplicial complex with vertex-decomposable dual is homotopy equivalent to a wedge of spheres.
Using the topological technique of diagrams of spaces, we calculate the homology of the union and the complement of finite arrangements of subspaces of the form $D + SP^{n-d}(X)$ in symmetric products $SP^n(X)$ where $D\in SP^d(X)$. As an application we include a computation of the homology of the homotopy end space of the open manifold $SP^n(M_{g,k})$, where $M_{g,k}$ is a Riemann surface of genus $g$ punctured at $k$ points, a problem which was originally motivated by the study of commutative $(m+k,m)$-groups.
This is a review paper about symmetric products of spaces $SP^n(X):= X^n/S_n$. We focus our attention on the symmetric products of 2-manifolds and make a journey through selected topics of algebraic topology, algebraic geometry, mathematical physics, theoretical mechanics etc. where these objects play an important role, demonstrating along the way the fundamental unity of diverse fields of physics and mathematics.
We express the signature ${\rm Sign}(SP^m_G(M))$ of the symmetric product $SP^n(M)$ of an (open) surface $M$ in terms of the cycle index $Z(G;\bar x)$ of $G$, a polynomial which originally appeared in P{\' o}lya enumeration theory of graphs, trees, chemical structures etc. The computations are used to show that there exist punctured Riemann surfaces $M_{g,k}, M_{g',k'}$ such that the manifolds $SP^{m}(M_{g,k})$ and $SP^{m}(M_{g',k'})$ are often not homeomorphic, although they always have the same homotopy type provided $2g+k = 2g'+k'$ and $k,k'\geq 1$.