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Register a new userOrientations and Connective Structures on 2-vector Bundles
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Authors
Thomas Kragh
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In work by Ausoni, Dundas and Rognes a half magnetic monopole is discovered and describes an obstruction to creating a determinant K(ku) to ku*. In fact it is an obstruction to creating a determinant gerbe map from K(ku) to K(Z,3). We describe this obstruction precisely using monoidal categories and define the notion of oriented 2-vector bundles, which removes this obstruction so that we can define a determinant gerbe. We also generalize Brylinskis notion of a connective structure to 2-vector bundles, in a way compatible with the determinant gerbe.
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We analyze the ring tmf_*tmf of cooperations for the connective spectrum of topological modular forms (at the prime 2) through a variety of perspectives: (1) the E_2-term of the Adams spectral sequence for tmf ^ tmf admits a decomposition in terms of Ext groups for bo-Brown-Gitler modules, (2) the image of tmf_*tmf in the rationalization of TMF_*TMF admits a description in terms of 2-variable modular forms, and (3) modulo v_2-torsion, tmf_*tmf injects into a certain product of copies of TMF_0(N)_*, for various values of N. We explain how these different perspectives are related, and leverage these relationships to give complete information on tmf_*tmf in low degrees. We reprove a result of Davis-Mahowald-Rezk, that a piece of tmf ^ tmf gives a connective cover of TMF_0(3), and show that another piece gives a connective cover of TMF_0(5). To help motivate our methods, we also review the existing work on bo_*bo, the ring of cooperations for (2-primary) connective K-theory, and in the process give some new perspectives on this classical subject matter.
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