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Symmetries and their anomalies give strong constraints on renormalizationgroup (RG) flows of quantum field theories. Recently, the identification of atheory's global symmetries with its topological sector has provided additionalconstraints on RG flows to symmetry preserving gapped phases due tomathematical results in category and topological quantum field theory. In thispaper, we derive constraints on RG flows from $\mathbb{Z}_2$-valued pure- andmixed-gravitational anomalies that can only be activated on non-spin manifolds.We show that such anomalies cannot be matched by a unitary, symmetry preservinggapped phase without symmetry fractionalization. In particular, we discussexamples that commonly arise in $4d$ gauge theories with fermions.

Anomaly Enforced Gaplessness and Symmetry Fractionalization for $Spin_G$ Symmetries