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Cobordism is a fundamental equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the boundary of a manifold.

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Naive G-spectrum representing geometric equivariant cobordism

Since my comment answered Emanuele Dotto's answer, I post it as an answer: Stefan Schwede discusses equivariant bordism in his book project about global homotopy theory in detail.
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Version of pseudo-isotopy $\neq$ isotopy for $(n+1)$-framings

Take any pseudoisotopy $\varphi\colon M\times I\rightarrow M\times I$ from the identity to a diffeomorphism $\phi$ that is not isotopic to the identity (as you mentioned, these exist). By obstruction …
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