• Corpus ID: 4978456

vn-ELEMENTS IN RING SPECTRA AND APPLICATIONS TO BORDISM THEORY

@inproceedings{Hovey1997vnELEMENTSIR,
  title={vn-ELEMENTS IN RING SPECTRA AND APPLICATIONS TO BORDISM THEORY},
  author={Mark Hovey},
  year={1997}
}

• Mark Hovey
• Published 1997
• Mathematics

The work of Hopkins and Smith [HS] has shown that the stable homotopy category has layered periodic behavior. On the (p-local) sphere, the only non-nilpotent selfmaps are multiplication by a power of p. But if we kill such a power to form the Moore space M(p), then we get a new family of non-nilpotent self maps, called the v1-self maps. Similarly, if we kill one of those, we get v2-self maps, and this behavior continues. One of the great advantages of the Brown-Peterson spectrum BP is that the periodicities are not layered, but they all appear as homotopy classes vn ∈ π2(pn−1)BP . Another great advantage of BP is that it is comparatively simple algebraically. Its coefficient ring is polynomial, and it is possible to calculate in the Adams-Novikov spectral sequence based on the operations in BP -homology. In fact, most of the spectra used by algebraic topologists are complex oriented, in that they admit maps from BP . But there is one crucial example that does not, namely, real K-theory KO. Hopkins and Miller [HMi] have recently shown that KO is the tip of an iceberg of non-complex oriented theories which have interesting torsion. It would be nice to have a bordism spectrum that did admit maps to the HopkinsMiller theories EOn. Recall that MO〈k〉 is the Thom spectrum arising from the k − 1-connected Postnikov cover BO〈k〉 of BO, and similarly for MU〈k〉. Note that MO〈4〉 = MSpin and MU〈4〉 = MSU both admit orientations to KO. I hope that this also is the beginning of a general phenomenon, and that the MO〈k〉 and MU〈k〉 will admit orientations to EOn when k is sufficiently large. Such an orientation may have some analytic meaning. Witten interprets a (conjectural) orientation from MO〈8〉 to elliptic cohomology, which should be EO2, as the index of an S-equivariant Dirac operator on the free loop space of a manifold with MO〈8〉 structure. However, it will be hard to get at the algebraic meaning of such an orientation because we know so little about the MO〈k〉. This paper is an attempt to get some qualititative understanding of the MO〈k〉 and MU〈k〉. We try to find analogues of vn in the homotopy of MO〈k〉 and MU〈k〉. It turns out to be easier to study the existence and properties of such analogues in