# Construction of exotic spheres that do not bound parallelizable manifolds

There are at least two ways to construct homotopy spheres that bound parallelizable manifolds, namely Milnor's plumbing construction and Brieskorn's method of singularities, and each of these methods has the virtue that it produces every element of $bP_{n+1}$ for every $n$.

What is the state of the art in constructing elements of $\Theta_n\setminus bP_{n+1}$?

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This seems to be a question that does not have a really satisfactory answer yet--at least to my knowledge--perhaps that is why nobody tried to answer it yet. Here is a not very satisfactory (because not really explicit) answer. Take a nonzero element $a$ in the cokernel of the stable J homomorfism. Choose a framed manifold representing an element in the stable homotopy groups of spheres, belonging to $a$. If the dimension of this manifold is odd, then by framed surgery one can kill all the homotopy groups below its dimension, and then you obtain an element in $\Theta_n -bP_{n+1}.$
This construction uses the isomorphism $Coker J \approx \Theta_n/bP_{n+1}.$
Here $J$ is the stable $J$-homomorphism $\pi_n(O) \to \pi_s(n).$