Enumerating Smooth Structures on $\mathbb{C}P^3\times\mathbb{S}^k$
Samik Basu, Ramesh Kasilingam, Ankur Sarkar
TL;DR
The paper addresses the problem of classifying smooth structures on products $\mathbb{C}P^3\times\mathbb{S}^k$ by computing concordance inertia groups and concordance structure sets via the Kirby–Siebenmann framework and stable-homotopy techniques. It derives a general decomposition $\mathcal{C}(M\times\mathbb{S}^k) \cong [M,Top/O] \oplus \pi_k(Top/O) \oplus [\Sigma^k M,Top/O]$, then carries out explicit computations for $M=\mathbb{C}P^3$ and $1\le k\le 10$, identifying inertia groups and providing a detailed diffeomorphism classification of all manifolds homeomorphic to $\mathbb{C}P^3\times\mathbb{S}^k$ for $1\le k\le 7$. The work combines deep stable-homotopy information, cofiber sequences, and surgery theory to produce concrete results such as $I(\mathbb{C}P^3\times\mathbb{S}^2)=0$, $I(\mathbb{C}P^3\times\mathbb{S}^3)=0$, $I(\mathbb{C}P^3\times\mathbb{S}^4)\cong \mathbb{Z}/3$, $I(\mathbb{C}P^3\times\mathbb{S}^5)\cong \mathbb{Z}/62$, and $I(\mathbb{C}P^3\times\mathbb{S}^7)=0$, together with explicit counts of diffeomorphism types in each case. This provides a complete picture of smooth structure variation on these CP^3–sphere products up to dimension seven and advances understanding of how higher exotic spheres influence diffeomorphism types via concordance invariants.
Abstract
In this paper, we compute the concordance inertia group of the product $M \times \mathbb{S}^k$, where $M$ is a simply connected, closed, smooth 6-manifold, for $1 \leq k \leq 10$, using known low-dimensional computations of the stable homotopy groups of spheres. Specifically, for $M = \mathbb{C}P^3$, we determine the inertia group of $\mathbb{C}P^3 \times \mathbb{S}^k$ for $2 \leq k \leq 7, k \neq 6$, and establish a diffeomorphism classification of all smooth manifolds homeomorphic to $\mathbb{C}P^3 \times \mathbb{S}^k$ for $1 \leq k \leq 7$.