All use subject to http://about.jstor.org/terms uation in which so many "textbook" techniques can be applied. The only pieces of background material that do lie outside the standard undergraduate curriculum are the principles from Morse theory alluded to earlier, so these are introduced and explained
In this paper, we continue the study of the homotopy type of spaces of rational functions from S2 to CPn begun in [3,4]. We prove that, for n > 1, Ratk(CPn) is homotopy equivalent to Ck(R2, S2n−1), the configuration space of distinct points in R2 with labels in S2n−1 of length at most k. This desuspends the stable homotopy theoretic theorems of [3, 4]. We also give direct homotopy equivalences between Ck(R2, S2n−1) and the Hilbert scheme moduli space for Ratk(CPn) defined by Atiyah and Hitchin [1]. When n = 1, these results no longer hold in general, and, as an illustration, we determine the homotopy types of Rat2(CP1) and C2(R2,S1) and show how they differ.
Abstract. In this paper, certain spectra B^k) are studied whose behavior qualifies them as being integral versions of the Brown-Gitler spectra B(k). The bulk of our work emphasizes the similarities between B^(k) and B(k), shown mainly using the techniques of Brown and Gitler. In analyzing the homotopy type of B^k), we provide a free resolution over the Steenrod algebra for its cohomology and study its Adams spectral sequence. We also list conditions which characterize it at the prime 2. The paper begins, however, on a somewhat different topic, namely, the construction of a configuration space model for Q2(S3(3)) and other related spaces.Introduction. In a paper published in 1973, E. H. Brown, Jr., and S. Gitler described a procedure by which one might conceivably find new characteristic classes for smooth rt-dimensional manifolds [3]. Their approach was based on examining the way that certain cohomology operations act on the stable normal bundle. Much of the motivation for their work stemmed from the problem of immersing manifolds in Euclidean space. For example, suppose that M is an n-manifold and let U denote the mod 2 Thorn class of its stable normal bundle. From standard properties of the Steenrod squares, it follows that if Sq' U ¥= 0, then M cannot be immersed in Ra+i~1.Brown and Gitler's idea was to take all those elements of the mod 2 Steenrod algebra which universally vanish on the Thorn classes of ^-manifolds and use these "primary" operations as the foundation for a coherent system of higher order cohomology operations which might then prove useful, say, to detect nonimmersions.
Abstract. In this paper, certain spectra B^k) are studied whose behavior qualifies them as being integral versions of the Brown-Gitler spectra B(k). The bulk of our work emphasizes the similarities between B^(k) and B(k), shown mainly using the techniques of Brown and Gitler. In analyzing the homotopy type of B^k), we provide a free resolution over the Steenrod algebra for its cohomology and study its Adams spectral sequence. We also list conditions which characterize it at the prime 2. The paper begins, however, on a somewhat different topic, namely, the construction of a configuration space model for Q2(S3(3)) and other related spaces.Introduction. In a paper published in 1973, E. H. Brown, Jr., and S. Gitler described a procedure by which one might conceivably find new characteristic classes for smooth rt-dimensional manifolds [3]. Their approach was based on examining the way that certain cohomology operations act on the stable normal bundle. Much of the motivation for their work stemmed from the problem of immersing manifolds in Euclidean space. For example, suppose that M is an n-manifold and let U denote the mod 2 Thorn class of its stable normal bundle. From standard properties of the Steenrod squares, it follows that if Sq' U ¥= 0, then M cannot be immersed in Ra+i~1.Brown and Gitler's idea was to take all those elements of the mod 2 Steenrod algebra which universally vanish on the Thorn classes of ^-manifolds and use these "primary" operations as the foundation for a coherent system of higher order cohomology operations which might then prove useful, say, to detect nonimmersions.
All use subject to http://about.jstor.org/terms uation in which so many "textbook" techniques can be applied. The only pieces of background material that do lie outside the standard undergraduate curriculum are the principles from Morse theory alluded to earlier, so these are introduced and explained
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