The $k=0$ and $k=1$ case are drawn up nicely in the second appendix of Freed and Uhlenbeck's classic book Instantons and Four-Manifolds. It's entitled the Pontrjagin-Thom Construction, and is motivated by wanting to prove a statement involving compute $[M,S^3]$ (for any compact simply-connected 4-manifold) whose nontriviality depends on the parity of the natural intersection formon compact simply-connected 4-manifolds.And the
The $k=0$ and $k=1$ case are drawn up nicely in the second appendix of Freed and Uhlenbeck's classic book Instantons and Four-Manifolds. It's entitled the Pontrjagin-Thom Construction, and is motivated by wanting to prove a statement involving the intersection forms form on general compact simply-connected manifolds4-manifolds. To boot, And the best part is that there is a cool picture of a dinosaur (that is, a cobordism) being cut open (that is, by two homotopy-equivalent framings).
The $k=0$ and $k=1$ case are drawn up nicely in the second appendix of Freed and Uhlenbeck's classic book Instantons and Four-Manifolds. It's entitled the Pontrjagin-Thom Construction, and is motivated by wanting to prove a statement involving intersection forms on general compact simply-connected manifolds. To boot, there is a cool picture of a dinosaur (that is, a cobordism) being cut open (by two homotopy-equivalent framings).