Stable cohomotopy in codimensions 2 and 3 receives complete algebraic characterizations for CW complexes and bordism interpretations for manifolds, yielding necessary and sufficient conditions for nowhere-vanishing vector bundle sections.
The Pontrjagin Dual of 3-Dimensional Spin Bordism
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For each space X we define an explicit group, G(X), functorially in X. This group is constructed from the groups of cochains on X. Furthermore, we construct an explicit functorial pairing with values in R/Z between the cochain representatives for elements of G(X) and maps of closed 3-dimensional spin manifolds to X. This pairing induces a pairing between G(X) and the 3-dimensional spin bordism group of X and identifies each with the Pontrjagin dual of the other.
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Stable Cohomotopy in Codimensions Two and Three: From Algebraic Characterizations to Bordism-Theoretic Interpretations
Stable cohomotopy in codimensions 2 and 3 receives complete algebraic characterizations for CW complexes and bordism interpretations for manifolds, yielding necessary and sufficient conditions for nowhere-vanishing vector bundle sections.