Abstract. A daisy graph is a union of immersed circles in 3-space which intersect only at the triple points. It is shown that a daisy graph can always be realized as the self-intersection set of an immersed closed surface in 3-space and the surface may be chosen to be orientable if and only if the daisy graph has an even number of edges on each immersed circle.It is well known that a general position immersion of a surface in euclidean 3-space may have two types of self-intersections: double points and triple points. If the surface is closed, then its self-intersection set is a union of immersed circles, called the transverse components, which intersect only at the triple points where three arcs meet in the same way as the three axes of 3-space meet at the origin. We call such a union of immersed circles a daisy graph which is so named since it can be obtained from copies of the daisy by adding bands; see Figure 1. For our purposes, we sometimes ignore the extra configurations at the triple points and consider a daisy graph as a union of embedded circles and a standard graph in 3-space with vertices being the triple points which are of degree 6, hence the notions of vertices and edges from graph theory make sense for a daisy graph and its transverse components. But observe that embedded circles that do not pass through triple points are considered to have zero number of edges.The study of immersed surfaces in 3-space via their self-intersections goes back as far as Dehn [8] and Whitney [12]. Banchoff [1] showed that the number of triple points of an immersed closed surface in 3-space is congruent modulo two to the Euler characteristic of the surface. The proof works in any 3-manifold provided the image of the surface is null-homologous. More generally, Carter and Ko proved a congruence χ(f (M )) = χ(M ) + T (f) modulo two, where χ denotes the Euler number of a cell complex, T (f ) is the number of triple points of an immersion f of a closed surface M into an arbitrary 3-manifold N ; see [4]. Their techniques could be applied to give a proof of the result of Izumiya and Marar [10] thatwhere B(f ) is the number of branched points, and now f is a general position map. This last result is an equality rather than a congruence. Carter [2] and Csikós and Szücs [7] addressed the problem of extending an immersion of a circle in a surface to a proper immersion of a surface in a 3-manifold bounded by the surface.

Abstract. We construct for each n ≥ 2 infinitely many n-component oriented links that are neither amphicheiral nor invertible; among these examples, infinitely many are Brunnian links.Amphicheirality and invertibility questions concerning knots and links in the 3-sphere S 3 are among the earliest problems in classical knot theory. These questions are of interest since they add to our knowledge of knot and link groups. The Jones polynomial and its various generalizations detect nonamphicheirality of a wide range of knots and links, but they fail to distinguish a single knot or link from its inverse.In this paper, we construct for each n ≥ 2 infinitely many n-component oriented links that are neither amphicheiral nor invertible; among these examples, infinitely many are Brunnian links. Our starting observation is that if an oriented link L in S It should be noticed that the existence of noninvertible knots was first established by Trotter [7]. Noninvertible links with an arbitrary number of components and invertible proper sublinks were first constructed by Whitten [8], [9]. However, Whitten's examples are not Brunnian links and the noninvertibility was established by difficult arguments from combinatorial group theory. Milnor'sμ-invariants [4], [5] of weight m have the ability to detect nonamphicheirality or noninvertibility of oriented links for m even or odd respectively. These invariants can be used to show that the n-component Brunnian link in Figure 7 of [4] is nonamphicheiral or noninvertible for n even or odd respectively. By using the algorithm provided by Cochran [1], further nonamphicheiral or noninvertible Brunnian links could be constructed. However, Milnor'sμ-invariants fail to detect noninvertibility of 2-component oriented links by Traldi [6] and it is not clear whether they detect noninvertibility of n-component oriented links for other even n.Definitions. An n-component oriented link in an oriented 3-manifold X is an ordered collection of n disjoint oriented circles tamely embedded in X. Two ncomponent oriented links L 1 and L 2 are equivalent or have the same link type if there exists an orientation preserving homeomorphism h of X onto itself such that

Let IΩn, k denote the bordism group of immersions of closed oriented n-manifolds into (n + k)-space. The object of this paper is to study certain group extension problems arising from Pastor's calculations of IΩn, k.The bordism group of immersions was first studied by Wells [12] who calculated the unoriented bordism group I Rn, k for k = n and k = n − 1 ≡ 3(4). Later these unoriented bordism groups were completely determined by Koschorke and Olk for k ≥ n − 2 with the help of an exact sequence measuring the difference between IRn, k and Rn (see [4]). A similar program has been carried out by Pastor [7] to determine the oriented bordism group I Ωn, k for k ≥ n − 2 except for certain group extension problems and some low dimensional cases.

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Abstract. Kojima's η-function is generalized to give a new concordance invariant for certain two-component manifold links in higher dimensions. Examples are given of manifold links successfully distinguished by this generalized η-function but not by their Cochran derived invariants.