Simply-connected manifolds with large homotopy stable classes

Abstract

For every k ≥ 2 and n ≥ 2 we construct n pairwise homotopically inequivalent simply-connected, closed 4k-dimensional manifolds, all of which are stably diffeomorphic to one another. Each of these manifolds has hyperbolic intersection form and is stably parallelisable. In dimension 4, we exhibit an analogous phenomenon for spin c structures on S 2 × S 2 .For m ≥ 1, we also provide similar (4m−1)-connected 8m-dimensional examples, where the number of homotopy types in a stable diffeomorphism class is related to the order of the image of the stable J-homomorphism π4m−1(SO) → π s 4m−1 .

Stably diffeomorphic manifolds and modified surgery obstructions

“…The primitive embeddings in Example 6.6 being distinct is the algebra underlying the constructions from our paper [CCPS21]. 6.2.…”

“…As a reality check, a brief calculation shows that λ M 1 = 2q a a −2s ⊕ (2q) does indeed vanish on D (for this calculation, it is helpful once again to recall that y 2 z 1 − y 1 z 2 = 1). This diagonal lagrangian was used implicitly in the construction of the manifolds N a,b from our previous article [CCPS21].…”

“…The simply-connected 4k-manifolds N a,b . Let us recall the next proposition from [CCPS21], which builds the simply-connected 4k-manifolds N a,b . We will then proceed to discuss some of their properties.…”

“…In [CCPS21] we gave the first examples of simply-connected 4k-manifolds, for k ≥ 2, with hyperbolic intersection form and arbitrarily large stable class. In contrast to Kreck and Schafer, we exhibited these examples in infinitely many stable classes.…”

“…that are pairwise not stably diffeomorphic. The manifolds N i that we constructed in [CCPS21] are recalled in Proposition 9.3, and then we define…”