Inertia groups of non-spin 4-dimensional complete intersections (Crowley–Nagy Conjecture 1.15)

An $n$-dimensional complete intersection $X_n(\underline d)$ is the smooth complex variety cut out by transverse hypersurfaces of degrees $d_1,\dots,d_k$ in $\mathbb{C}P^{n+k}$. Crowley and Nagy proved the 4-dimensional Sullivan Conjecture — complete intersections of complex dimension 4 with the same total degree, Pontryagin numbers, and Euler characteristic are diffeomorphic — and determined the inertia group $I(X_4(\underline d))\subseteq\Theta_8\cong\mathbb{Z}/2$ ($\Theta_8$ the Kervaire–Milnor group of homotopy 8-spheres) in the spin case and in the non-spin case with vanishing fourth Wu class. For the remaining non-spin complete intersections with $v_4(X_4(\underline d))\neq 0$, they conjecture (Conjecture 1.15, Geometry & Topology 29 (2025), 269–311) that $I(X_4(\underline d))=0$ when $p_1(4,\underline d)\equiv 3\pmod 8$ and $I(X_4(\underline d))=\Theta_8$ when $p_1(4,\underline d)\equiv 7\pmod 8$. The problem is to prove or refute these two predicted rows, which together predict, for every multidegree $\underline d$ with $X_4(\underline d)$ non-spin and $v_4\neq 0$, whether $X_4(\underline d)$ is diffeomorphic to its connected sum with the exotic 8-sphere. The conjecture is open as stated in the published version.

Background

A complete intersection $X_n(\underline d)$ is a closed, simply connected $2n$-manifold obtained as the transverse intersection in $\mathbb{C}P^{n+k}$ of $k$ hypersurfaces of degrees $d_1,\dots,d_k$; the product $d=d_1\cdots d_k$ is the total degree. Sullivan, in his 1970 MIT notes, asked whether such manifolds are classified up to diffeomorphism by their Sullivan data $SD_n(\underline d)=(d,(p_i(n,\underline d))_{i=1}^{\lfloor n/2\rfloor},\chi)$, where $p_i(n,\underline d)$ is the integer defined by $p_i(X_n(\underline d))=p_i(n,\underline d)\,x^{2i}$ for the standard generator $x\in H^2$ pulled back from projective space (in the convention of Crowley–Nagy, $p(X_n(\underline d))=(1-x^2)^{n+k+1}\prod_{i=1}^{k}(1-d_i^2x^2)^{-1}$, so that $p_1(4,\underline d)=\sum_{i=1}^{k}d_i^2-(k+5)$ and e.g. $p_1(\mathbb{C}P^4)=-5$), and $\chi$ is the Euler characteristic. Libgober and Wood used this classification to count connected components of complex-moduli spaces of complete intersections, which is one main motivation. Prior partial results: for $n=4$, Fang and Klaus (Manuscripta Math. 90 (1996)) proved the conjecture up to connected sum with a homotopy 8-sphere, and Kreck (Surgery and Duality, 1999), in work with Traving, proved it when 64 divides the total degree. By Kervaire and Milnor's computation (Ann. of Math. 77 (1963)), the group $\Theta_8$ of homotopy 8-spheres up to oriented diffeomorphism is $\mathbb{Z}/2$; write $\Sigma^8_{ex}$ for the exotic generator. The inertia group of a closed manifold $M$ is $I(M)=\{\Sigma\in\Theta_8:\;M\cong M\sharp\Sigma\}$; $M$ is called $\Theta$-rigid when $I(M)=\Theta_8$ and $\Theta$-flexible when $I(M)=0$. Thus the Fang–Klaus indeterminacy is exactly the question of whether a given $X_4(\underline d)$ is $\Theta$-rigid. Crowley and Nagy (Geom. Topol. 29 (2025) 269–311) proved the 4-dimensional Sullivan Conjecture in all cases, and their analysis splits the inertia question by the Wu classes $v_2,v_4\in\mathbb{Z}/2$ (the Wu classes are the cohomology classes satisfying $Sq(v)=w$ for the Stiefel–Whitney classes; by Remark 2.6 of the paper they are single $\mathbb{Z}/2$-valued bits for these manifolds, with $v_2=w_2$ detecting spin): (i) if $X_4(\underline d)$ is spin, the total space $S(X_4(\underline d),\pm x)$ of the circle bundle with Chern class $\pm x$ is framed null-cobordant while $S(X_4(\underline d)\sharp\Sigma^8_{ex},\pm x)$ is not, giving a geometric diffeomorphism invariant, so $X_4(\underline d)$ is $\Theta$-flexible with $I=0$ (Theorem 1.7); (ii) if $X_4(\underline d)$ is non-spin with $v_4=0$, the manifold is $\Theta$-rigid, $I=\Theta_8$ (Theorem 1.12). The non-spin case with $v_4\neq 0$ — which forces $p_1(4,\underline d)\equiv 3\pmod 4$, so that $p_1(4,\underline d)\bmod 8$ is either 3 or 7 — remains undecided in general, even though the Sullivan Conjecture itself is proved in these cases too (via Kreck's modified surgery: a normal 3-smoothing is a 3-connected bundle-covered map to the normal type $(B_4,\xi_4(\underline d)\times\gamma_{BO\langle 8\rangle})$, with $\xi_4(\underline d)$ the stable normal bundle over $\mathbb{C}P^\infty$; bordant smoothings plus equal Euler characteristic imply diffeomorphism). The rigidity question in these families reduces to computing the canonical homomorphism $i_0:\Theta_8\cong\mathrm{Tors}\,\Omega^{O\langle 7\rangle}_8\to\Omega^{O\langle 7\rangle}_8(\mathbb{C}P^\infty;\xi_4(\underline d))$ into the twisted string bordism group ($O\langle 7\rangle$ is the 7-connected cover of $O$, i.e. the string structure on the stable normal bundle), which the authors analyse through the homotopy type of the Thom spectrum of $\xi_4(\underline d)|_{\mathbb{C}P^4}$; $i_0=0$ forces $\Theta$-rigidity, while injectivity is expected to be obstructive. They note (after Conjecture 1.15) that in the $\Theta$-flexible non-spin cases no purely geometric diffeomorphism invariant analogous to the spin-case circle-bundle framing property is known. Kasilingam (Proc. Indian Acad. Sci. Math. Sci. 126 (2016)) showed $I(\mathbb{C}P^4)=0$. The inertia-group result of Senger–Zhang does not apply here, because complete intersections have $\pi_2\cong\mathbb{Z}$.

Problem Statement

Determine the inertia group $I(X_4(\underline d))\subseteq\Theta_8\cong\mathbb{Z}/2$ of the 4-dimensional complete intersection $X_4(\underline d)$ for every multidegree $\underline d$ such that $X_4(\underline d)$ is non-spin and $v_4(X_4(\underline d))\neq 0$, where $v_2,v_4$ are the Wu classes and $p_1(4,\underline d)$ is defined by $p_1(X_4(\underline d))=p_1(4,\underline d)\,x^2$ for the standard generator $x\in H^2(X_4(\underline d))$ (with the sign convention $p(X_4(\underline d))=(1-x^2)^{k+5}\prod_{i=1}^{k}(1-d_i^2x^2)^{-1}$ fixed in Background, i.e. $p_1(4,\underline d)=\sum_{i=1}^{k}d_i^2-(k+5)$). Specifically, settle each of the two open rows of Crowley–Nagy's Conjecture 1.15, independently, by proving or refuting it: (a) if $p_1(4,\underline d)\equiv 3\pmod 8$ then $I(X_4(\underline d))=0$ (i.e. $X_4(\underline d)\sharp\Sigma^8_{ex}$ is not diffeomorphic to $X_4(\underline d)$ for the exotic 8-sphere $\Sigma^8_{ex}$), and (b) if $p_1(4,\underline d)\equiv 7\pmod 8$ then $I(X_4(\underline d))=\Theta_8$ (i.e. $X_4(\underline d)$ is $\Theta$-rigid). Each row is a universal claim over the qualifying multidegrees with the given residue of $p_1(4,\underline d)\bmod 8$: proving a row determines $I(X_4(\underline d))$ — equivalently, whether $X_4(\underline d)$ is diffeomorphic to $X_4(\underline d)\sharp\Sigma^8_{ex}$ — for every such multidegree, while refuting a row requires only one qualifying multidegree in that class whose inertia group is opposite to the row's prediction, and it settles that row alone, leaving $I(X_4(\underline d))$ undetermined for the remaining multidegrees of the class. The two rows are decided separately; proving both determines $I(X_4(\underline d))$ for every qualifying multidegree.

References

1. Diarmuid Crowley and Csaba Nagy, The smooth classification of 4-dimensional complete intersections, Geometry & Topology 29 (2025) 269–311, DOI 10.2140/gt.2025.29.269, arXiv:2003.09216, https://arxiv.org/abs/2003.09216
2. Fuquan Fang and Stephan Klaus, Topological classification of 4-dimensional complete intersections, Manuscripta Mathematica 90 (1996) 139–147, DOI 10.1007/BF02568299, https://doi.org/10.1007/BF02568299
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