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Suppose M 1 {\displaystyle M_{1}} and M 2 {\displaystyle M_{2}} are framed submanifolds of a differential manifold P {\displaystyle P} . A framed cobordism between M 1 {\displaystyle M_{1}} and M 2 {\displaystyle M_{2}} in P {\displaystyle P} is a framed submanifold with boundary ( N , ∂ N ) {\displaystyle (N,\partial N)} inside P × I {\displaystyle P\times I} , that provides a cobordism between M 1 × { 0 } {\displaystyle M_{1}\times \{0\}} and M 2 × { 1 } {\displaystyle M_{2}\times \{1\}} .
