To a plane algebraic curve of degree n, Moishezon associated a braid monodromy homomorphism from a finitely generated free group to Artin's braid group B_n. Using Hansen's polynomial covering space theory, we give a new interpretation of this construction. Next, we provide an explicit description of the braid monodromy of an arrangement of complex affine hyperplanes, by means of an associated ``braided wiring diagram.'' The ensuing presentation of

In this paper we compute the second homology of the discrete jet groups. The n-th jet group, J_n, is the group, under composition followed by truncation, of invertible, orientation-preserving real n-jets at 0. Consider the homomorphism D: J_n o R^+ obtained by projecting onto the first coefficient. The main result of this paper is: The map D_*: H_2(J_n) o H_2(R^+) is an isomorphism.

We use covering space theory and homology with local coefficients to study the Milnor fiber of a homogeneous polynomial. These techniques are applied in the context of hyperplane arrangements, yielding an explicit algorithm for computing the Betti numbers of the Milnor fiber of an arbitrary real central arrangement in C^3, as well as the dimensions of the eigenspaces of the algebraic monodromy. We also obtain combinatorial formulas for these invariants

The Chen groups of a group are the lower central series quotients of its maximal metabelian quotient. We show that the Chen groups of the pure braid group P_n are free abelian, and we compute their ranks. The computation of these Chen groups reduces to the computation of the Hilbert series of a certain graded module over a polynomial ring, and the latter is carried out by means of a Groebner basis algorithm. This result shows that, for n ge 4, the

Consider a split exact sequence of discrete groups $${1}to G to Gamma{underset sigma to {overset{pi} to rightleftarrows}} Gamma/G to {1}.$$ Suppose there exists a normal series $G=G_0 triangleright G_1 triangleright cdots triangleright G_n triangleright G_{n+1}={1}$, such that (1) $G_i/G_{i+1}$ is a rational vector space for $i=0, cdots, n$; (2) $G_i/G_{i+1}$ is contained in the center of $G/G_{i+1}$ for $i=0, cdots, n$; (3) there exists an element

Given a closed $n$-manifold $M^n$ and a tuple of positive integers $P$, let $sigma_P M$ be the $P$-spin of $M$. If $M otsimeq S^n$ and $P e Q$ (as unordered tuples), it is shown that $sigma_P M otsimeq sigma_Q M$ if either (1) $H_*(M) otcong H_*(S^n)$, (2) $pi_1(M)$ is finite, (3) $M$ is aspherical, or (4) $n=3$. Applications to the homotopy classification of homology spheres and knot exteriors are given.

We show that the oriented homotopy type of a spun 3-manifold is determined by the fundamental group and the choice of framing.

If M is a compact, connected, simply-connected, smooth 4-manifold, and gamma is a class in H_2(M; Z), define d_gamma to be the minimum number of double points of immersed spheres representing gamma. We use a theorem of S. K. Donaldson to provide lower bounds for d_gamma, for gamma certain homology classes in rational surfaces.