The Sullivan conjecture for complete intersections predicts that two smooth complex projective complete intersections of the same complex dimension n≥3n\ge3 with the same total degree, the same Pontryagin classes and the same Euler characteristic must be diffeomorphic, i.e. that this Sullivan data is a complete smooth invariant. The conjecture is proved for n=3n=3 (classification of simply-connected 6-manifolds) and n=4n=4 (Crowley–Nagy 2025), is known up to homeomorphism for 5≤n≤75\le n\le7 (Fang–Wang), and holds in all dimensions only under strong divisibility conditions on the total degree (Kreck, following Traving). Baraglia's computation of the alpha invariant (the mod-2 index of the Dirac operator) of spin complete intersections of dimension n≡1(mod4)n\equiv1\pmod4, n≥5n\ge5, shows that the most natural potential counterexample invariant is in fact determined by the Sullivan data, adding consistency evidence but leaving the conjecture open. This problem asks for a resolution of the remaining open cases: a proof that equal Sullivan data implies diffeomorphism for all complex dimensions n≥5n\ge5 (where for 5≤n≤75\le n\le7 this means upgrading the known homeomorphism classification, and for n≥8n\ge8 even the topological classification is open without degree divisibility), or an explicit pair of complete intersections with identical Sullivan data that are not diffeomorphic.

Background

A smooth complex projective complete intersection Xn(d1,…,dk)X_n(d_1,\dots,d_k) is the transverse intersection of kk nonsingular complex hypersurfaces of degrees d1,…,dkd_1,\dots,d_k inside the complex projective space CPn+k\mathbb{CP}^{n+k}; it is a closed complex nn-dimensional (real 2n2n-dimensional) manifold, simply-connected for n≥2n\ge2, and a fundamental testing ground for the diffeomorphism classification of algebraic varieties. A classical theorem of René Thom (1923–2002) states that the diffeomorphism type of the underlying smooth 2n2n-manifold depends only on the dimension nn and the multi-degree (d1,…,dk)(d_1,\dots,d_k) up to reordering and inserting or removing entries equal to 1. By the Lefschetz hyperplane theorem (which says a hypersurface section induces isomorphisms on cohomology below the middle dimension), for n≥3n\ge3 the second cohomology H2(Xn(d‾);Z)H^2(X_n(\underline d);\mathbb Z) is infinite cyclic, generated by a class xx. Consequently each Pontryagin class pip_i — a characteristic class of the tangent bundle measuring, roughly, signed curvature contributions, with rational coefficients expressible in terms of Chern classes — satisfies pi=pi(n,d‾) x2ip_i = p_i(n,\underline d)\,x^{2i} for an explicit integer pi(n,d‾)p_i(n,\underline d), and xnx^n equals dtot=d1d2⋯dkd_{\mathrm{tot}}=d_1d_2\cdots d_k (the total degree) times a generator of top cohomology, so the Euler characteristic is likewise an explicit function of nn and d‾\underline d. The tuple of total degree, integer Pontryagin classes, and Euler characteristic is called the Sullivan data SDn(d‾)SD_n(\underline d); it is computable from the multi-degree via the total Chern class formula

c(Xn(d‾))=(1+x)n+k+1∏j=1k(1+djx)−1.c(X_n(\underline d)) = (1+x)^{n+k+1}\prod_{j=1}^{k}(1+d_j x)^{-1}.

For n≥3n\ge3, diffeomorphic complete intersections necessarily have equal Sullivan data. Dennis Sullivan conjectured in the 1970s that the converse holds, and this Sullivan conjecture for complete intersections has driven the field since: if true, the Sullivan data is a complete diffeomorphism invariant of complete intersections. The confirmed cases are: n=3n=3, by Wall's and Jupp's classification of simply-connected 6-manifolds; n=4n=4 smoothly, proved by Crowley and Nagy (published 2025), building on Fang and Klaus's earlier result that the n=4n=4 case holds up to connected sum with a homotopy 8-sphere (a homotopy sphere is a smooth manifold homeomorphic but possibly not diffeomorphic to the standard sphere; the group Θm\Theta_m of their diffeomorphism classes is finite by Kervaire–Milnor); and 5≤n≤75\le n\le7 up to homeomorphism, by Fang and Wang. In arbitrary dimension n≥3n\ge3, Kreck's modified surgery theory (a refinement of surgery theory that classifies manifolds through their normal bundle data in place of homotopy type; building on Traving's 1985 thesis) proves the conjecture under a strong hypothesis: the total degree must be divisible by a high power of each small prime, e.g. by 64 when n=4n=4. One route to a counterexample would be a diffeomorphism invariant not determined by the Sullivan data. A spin manifold is one whose tangent bundle admits a spin structure, possible precisely when the first two Stiefel–Whitney classes vanish; a spin complete intersection of dimension n≥2n\ge2 has a unique spin structure, and then one can define its alpha invariant α∈KO−n(pt)\alpha\in KO^{-n}(\mathrm{pt}), the refined A^\hat A-genus given by the index of the Dirac operator. For n≡1(mod4)n\equiv1\pmod4 this is the mod-2 index, an element of Z/2\mathbb Z/2 that a priori need not be determined by the Pontryagin classes, making it the natural candidate obstruction. Baraglia (2020) computed this invariant for all spin complete intersections of dimension n≡1(mod4)n\equiv1\pmod4, n≥5n\ge5, and proved that it depends only on the total degree and Pontryagin classes, so the alpha invariant provides no counterexample; Crowley and Nagy use exactly this to control the Z/2\mathbb Z/2-factor of Θ10≅Z/2×Z/3\Theta_{10}\cong\mathbb Z/2\times\mathbb Z/3 in their announced program for dimension 5. A further motivation is the result of Libgober and Wood that diffeomorphic complete intersections with different multi-degrees carry complex structures in different connected components of the moduli space of complex structures, so the conjecture directly governs how such components can be counted.

Problem Statement

Resolve the Sullivan conjecture for complete intersections in its remaining open cases. The conjecture: for n≥3n\ge3, any two nn-dimensional smooth complex projective complete intersections Xn(d‾)X_n(\underline d) and Xn(d‾′)X_n(\underline d') whose Sullivan data agree — the same total degree dtot=∏jdjd_{\mathrm{tot}}=\prod_j d_j, the same Pontryagin classes pip_i, and the same Euler characteristic — are diffeomorphic. As in the standard formulation of Crowley and Nagy, each complete intersection carries its complex orientation and diffeomorphism is understood orientation-preserving. It is proved for n=3n=3 and n=4n=4; the open target is the conjecture for every complex dimension n≥5n\ge5 (for 5≤n≤75\le n\le7 this amounts to upgrading Fang–Wang's homeomorphism classification to diffeomorphism, controlling the action of the finite group Θ2n\Theta_{2n} of homotopy spheres; for n≥8n\ge8 no unconditional classification theorem of either kind is known). The conjecture would be resolved by a proof that equal Sullivan data implies diffeomorphism for all complete intersections of each complex dimension n≥5n\ge5, or by a counterexample: for some n≥5n\ge5, an explicit pair of multi-degrees d‾,d‾′\underline d,\underline d' whose complete intersections have identical Sullivan data but are not diffeomorphic.

References

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