Low-dimensional G-bordism and G-modular TQFTs

Let G denote a class of manifolds (such as SO (oriented), O (unoriented), Spin, Pin+, Pin-, manifolds with spin defects).  We define a 2+1-dimensional G-modular TQFT to be one which lives on the boundary of a bordism-invariant 3+1-dimensional G-TQFT.  Correspondingly, we define a G-modular braided category to be a G-premodular category which leads to a bordism-invariant 3+1-dimensional TQFT.  When G = SO, this reproduces the familiar Witten-Reshetikhin-Turaev TQFTs and corresponding modular tensor categories.  For other examples of G, non-zero G-bordism groups in dimensions 4 or lower lead to interesting complications (anomalies, mapping class group extensions, obstructions to defining the G-modular theory on all G-manifolds).

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