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 $\Theta_n\setminus bP_{n+1}$?


I don't really know about the state of the art, but I've come across a couple of examples in the literature at least. In the article:

Frank, David L., An invariant for almost-closed manifolds, Bull. Amer. Math. Soc. 74 (1968) 562–567, MR0222906

the unique exotic $8$-sphere and an order-$3$ element of $\Theta_{10} \cong \mathbb{Z}/6$ are shown to be in the image of Milnor's plumbing construction; see Examples 1 and 2 on page 565. Since they're exotic (i.e. not the standard sphere) and even-dimensional, they do not bound any parallelisable manifold (since $bP_{2n+1} = \{S^{2n}\}$). Also, in the article:

Sperança, L. D., Pulling back the Gromoll-Meyer construction and models of exotic spheres, Proc. Amer. Math. Soc. 144 (2016), no. 7, 3181–3196, MR3487247

there is an explicit description of clutching diffeomorphisms that realise these two examples as twisted spheres; see Theorem 4.6 (plus the description in the middle of page 3187 of "reentrance").


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 $\pi_n(O) \to \pi_s(n).$


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