On the state of the art on closed $(n-1)$-connected $2n$ manifolds

Question

In the paper "Classification of $(n - 1)$-Connected $2n$-Manifolds" by C.T.C.Wall (Annals of Mathematics , Jan., 1962, Second Series, Vol. 75, No. 1 (Jan., 1962), pp. 163-189), Wall studies $(n - 1)$-Connected $2n$-Manifolds with a small ball removed and proves a classification result for such manifolds in terms of algebro-topological invariants, namely the intersection form on the middle cohomology and a homotopy theoretic invariant which varies finitely.

In the introduction Wall makes the following remark (I quote): "In a subsequent paper, the author intends to study the diffeomorphisms of the manifolds here obtained; in particular, to give a complete set of invariants of isotopy of a diffeomorphism, and to consider more carefully the actual diffeomorphism classification of closed $(n-1)$-connected $2n$-manifolds (which is not settled in this paper, even when our results are complete.)"

There are two parts to my question:

1. Did the mentioned paper appear?

I should mention that I have looked through the titles of Wall's (100+) subsequent articles and not found a title directly related to this problem, hence the question.

I should also mention what is known in low dimensions. For $2n = 4$ this includes the smooth Poincare conjecture so is open to my knowledge, up to homeomorphism it has been settled by Freedman. In dimension $2n = 6$, I believe that it is known due to work of Wall, Jupp and Zubr that such a manifold should be diffeomorphic to a connect sum of $S^3 \times S^3$'s.

I ask a second part:

2. Let $n$ be an even integer, $n \geq 4$. What is the state of the art of the homoemorphism classification of closed $(n-1)$-connected $2n$-manifolds?

Note also in Question 2 I have intentionally slightly modified the problem, since manifolds with dimension $2 \mod 4$ and the issue of different smooth structures are not of primary interest to me.

Answer 1

The classification problem of smooth oriented closed $(n-1)$-connected $2n$-manifolds for $n\ge3$ splits into three parts.

1. Classify smooth almost closed compact oriented $(n-1)$-connected $2n$-manifolds, where almost closed means that the boundary is a homotopy sphere.
2. Understand those homotopy spheres that arise as boundaries of almost closed compact oriented smooth $(n-1)$-connected $2n$-manifolds.
3. Understand inertia groups of smooth oriented closed $(n-1)$-connected $2n$-manifolds.

In the work you mentioned, Wall achieved a classification of type 1. in terms of what he calls n-forms. Since then several authors (Wall, Kosinski, Schultz, Stolz, ...) have obtain partial results regarding 2. and 3. Most recently Burklund--Hahn--Senger and Burklund--Senger settled the last open cases and completed the classification.

Regarding your second question: In the topological category, 2. and 3. are vacuous since all high-dimensional homotopy spheres are topologically trivial. Wall's original approach to 1. only uses tools that have since been established in the topological category (mostly by Kirby--Siebenmann), so his approach goes through and reduces the classification to understanding $\pi_n(BSTop(n))$. To get at the latter, you can compare $BSTop(n)$ to $BSTop$ and use that its homotopy fibre receives a highly-connected map from $SO/SO(n)$.