Narrow Results

We extend the vanishing and sphere theorems due to Lawson, Simons, Xin, Shiohama and Xu. By using the techniques of calculus of variations in the geometric measure theory, we prove the vanishing theorem for homology groups of submanifolds in the hyperbolic space Hⁿ(c) with negative constant curvature c. Moreover, we obtain a topological sphere theorem for certain complete submanifolds in Hⁿ(c).

There is a simple group-theoretic formula for the second integral homology group of a group. This is an abelian group and there is an analogous formula for another abelian group, which involves a normal subgroup N of a torsion-free nilpotent group G. Properties of this abelian group translate into properties of G/N. This approach allows one to give a simple purely group-theoretic proof of an old theorem of J. R. Stallings, namely that if Γ is a group, if H₁(G,ℤ) is free abelian and if H₂(G,ℤ) = 0, then any subset Y of G which is independent modulo the derived group of G, freely generates a free group. The ideas used admit to considerable generalization, yielding in particular, proofs of a number of theorems of U. Stammbach.

Fox showed that the order of homology of a cyclic branched cover of a knot is determined by its Alexander polynomial. We find examples of knots with relatively prime Alexander polynomials such that the first homology groups of their q-fold cyclic branched covers are of the same order for every prime power q. Furthermore, we show that these knots are linearly independent in the knot concordance group using the polynomial splitting property of the Casson–Gordon–Gilmer invariants.

A quandle is a set equipped with a binary operation satisfying three quandle axioms. It also can be expressed as a sequence of permutations of the underlying set satisfying certain conditions. In this paper, we will calculate the second rack homology group of the disjoint union of two finite quandles and the second and third rack homology groups of certain type of finite quandles.

In this paper, we study closed orientable Euclidean manifolds which are also known as flat three-dimensional manifolds or just Euclidean 3-forms. Up to homeomorphism, there are six of them. The first one is the three-dimensional torus. In 1972, Fox showed that the 3-torus is not a double branched covering of the 3-sphere. So, it is not a hyperelliptic manifold. In this paper, we show that all the remaining Euclidean 3-forms are hyperelliptic manifolds.

We obtain a necessary condition for homology group to be zero on CR-warped product submanifold in Euclidean spaces in terms of second fundamental form of the submanifold and warping function. By using this condition, we show that such CR-warped product submanifold is a homotopy sphere.