= Cobordism hypothesis
The <cobordism hypothesis> classifies fully extended framed <topological quantum field theories> by the object assigned to a point. Precisely, for a suitable symmetric monoidal $(\infty,n)$-category $\mathcal C$, evaluating a framed theory at a positively framed point identifies its space of theories with the space of <fully dualizable objects> of $\mathcal C$. The word “fully” records duals and adjoints at the lower-dimensional levels needed to cut and glue manifolds.<On the Classification of Topological Field Theories by Jacob Lurie>[Section 2, formulation and <fully dualizable objects>]{ref}
In one dimension, a finite-dimensional <vector space> $V$ has a dual $V^*$. Evaluation and coevaluation describe interval pieces; closing them into a circle gives the trace of the identity, $Z(S^1)=\dim V$. This is the simplest glimpse of how assigning data to a point controls closed-manifold invariants.
The framing hypothesis matters. Passing to oriented or other tangentially structured theories requires corresponding extra compatibility data, often expressed as a homotopy-fixed-point condition. Nor does the statement classify arbitrary unextended field theories by their point values: those theories do not include all the lower-dimensional data in the first place.
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