We write down the chain complex C∗ of the universal cover of the torus, and indicate how to do likewise for any 2-dimensional CW complex arising as the 2-skeleton of a K(G, 1). We exhibit a contracting chain homotopy for this chain complex, and again explain how to do so for the chain complex of the universal cover of our more general CW complex. This yields a contracting homotopy for the tenso...

let $gamma(s_n)$ be the minimum number of proper subgroups‎ ‎$h_i, i=1‎, ‎dots‎, ‎l $ of the symmetric group $s_n$ such that each element in $s_n$‎ ‎lies in some conjugate of one of the $h_i.$ in this paper we‎ ‎conjecture that $$gamma(s_n)=frac{n}{2}left(1-frac{1}{p_1}right)‎ ‎left(1-frac{1}{p_2}right)+2,$$ where $p_1,p_2$ are the two smallest primes‎ ‎in the factorization of $ninmathbb{n}$ an...

In this paper, we propose a new type of matroids, namely covering matroids, and investigate the connections with the second type of covering-based rough sets and some existing special matroids. Firstly, as an extension of partitions, coverings are more natural combinatorial objects and can sometimes be more efficient to deal with problems in the real world. Through extending partitions to cover...

We develop an explicit covering theory for complexes of groups, parallel to that developed for graphs of groups by Bass. Given a covering of developable complexes of groups, we construct the induced monomorphism of fundamental groups and isometry of universal covers. We characterize faithful complexes of groups and prove a conjugacy theorem for groups acting freely on polyhedral complexes. We a...

We study the existence of geometrically controlled branched covering maps from R to open 3-manifolds or to decomposition spaces S/G, and from S/G to S. Dedicated to the memory of Fred Gehring, whose work has been our inspiration

In this paper we define, for each aspherical orientable 3-manifold M endowed with a torus splitting T , a 2-dimensional fundamental l1-class [M ] whose l1norm has similar properties as the Gromov simplicial volume of M (additivity under torus splittings and isometry under finite covering maps). Next, we use the Gromov simplicial volume of M and the l1-norm of [M ] to give a complete characteriz...

The main theorem of this paper states that ifM is a closed orientable hyperbolic 3-manifold of volume at most 3.08, then the dimension of H1(M ;Z2) is at most 7, and that it is at most 6 unless M is “strange.” To say that a closed, orientable 3-manifold M , for which H1(M ;Z2) has dimension 7, is strange means that the Z2-vector space H1(M ;Z2) has a 2-dimensional subspace X such that for every...

For a set of n points in the plane, we consider the axis–aligned (p, k)-Box Covering problem: Find p axis-aligned, pairwise-disjoint boxes that together contain at least n − k points. In this paper, we consider the boxes to be either squares or rectangles, and we want to minimize the area of the largest box. For general p we show that the problem is NP-hard for both squares and rectangles. For ...

Let A be a closed subgroup of a connected, solvable Lie group G, such that the homogeneous space A\G is simply connected. As a special case of a theorem of C. T. C. Wall, it is known that every tessellation A\G/Γ of A\G is finitely covered by a compact homogeneous space G′/Γ′. We prove that the covering map can be taken to be very well behaved — a “crossed” affine map. This establishes a connec...

We notice that there is a 2-cover P of S 1 SO(3) (we denote this group D 1/2 ) by the group U(2): P sends a matrix z into a pair (det z,v). The matrix v here is the image of u under a standard covering map p from SU(2) onto SO(3), see (3.4) below. Finally, u (being a matrix from SU(2) ) is determined (up to a sign) from the decomposition z = du, here d 2 = det z. Both P and p are group homomorp...