For ordinary complete intersections in complex projective space, the Sullivan conjecture predicts that for complex dimension at least 3 the diffeomorphism type is determined by three classical invariants: total degree, Pontrjagin classes, and Euler characteristic. Wang and Wang (2012) proved the weighted analogue of the Kreck-type case of this criterion for smooth weighted complete intersections of odd complex dimension, with pairwise coprime odd weights, under a mod-p Adams filtration divisibility bound, and for pairs sharing a common smooth ambient weighted hypersurface. Whether the criterion survives outside this regime — even complex dimension, weights that are not all odd or not pairwise coprime, multidegrees violating the divisibility bound, or more general ambient setups — is unresolved. The problem asks for a proof of the Sullivan-type criterion in any of these regimes, or an explicit counterexample pair with identical invariants that is not diffeomorphic.

Background

A smooth complete intersection Xn(d1,…,dk)X_n(d_1,\dots,d_k) is the transverse common zero locus of kk homogeneous polynomials of degrees d1,…,dkd_1,\dots,d_k in CPn+k\mathbb{CP}^{n+k}; it is a closed oriented smooth manifold of real dimension 2n2n. The classical organizing problem of their differential topology, emerging from Sullivan's infinitesimal-computations machinery (Sullivan 1977) and the work of Libgober and Wood (1982), is the Sullivan conjecture: for complex dimension n≥3n\geq 3, two complete intersections are orientation-preserving diffeomorphic if and only if they have the same total degree d=d1⋯dkd=d_1\cdots d_k (the product of the degrees), the same Pontrjagin classes, and the same Euler characteristic. Because the Pontrjagin classes of such a manifold are multiples of powers of the hyperplane class, these data reduce to a finite tuple of integers, called the Sullivan data. Writing νp(m)\nu_p(m) for the exponent of a prime pp in the factorization of mm, the known history is: dimension 3 follows from classical classification theorems for simply-connected 6-manifolds (see the account in Crowley and Nagy 2025); in dimension 4, Fang and Klaus (1996) proved the conjecture up to connected sum with homotopy 8-spheres, and Kreck (1999, Theorem A of Surgery and duality) proved it in all dimensions n≥3n\geq 3 under the arithmetic bound νp(d)≥2n+12(p−1)+1\nu_p(d)\geq\frac{2n+1}{2(p-1)}+1 for every prime pp with p(p−1)≤n+1p(p-1)\leq n+1 (for n=4n=4 this says 6464 divides dd); Fang and Wang (2010) settled the homeomorphism classification for 5≤n≤75\leq n\leq 7; and Crowley and Nagy (2025) removed the arithmetic restriction in dimension 4, completing the 4-dimensional Sullivan conjecture using the theory of degree-dd normal maps of Hambleton and Madsen (1986).

A weighted projective space P(ω0,…,ωN)\mathbb{P}(\omega_0,\dots,\omega_N) is the quotient of CN+1∖{0}\mathbb{C}^{N+1}\setminus\{0\} by the weighted scalar action λ⋅(z0,…,zN)=(λω0z0,…,λωNzN)\lambda\cdot(z_0,\dots,z_N)=(\lambda^{\omega_0}z_0,\dots,\lambda^{\omega_N}z_N); for general weights it has cyclic quotient singularities, and its integral cohomology, computed by Kawasaki (1973), can contain torsion — 2-torsion appears as soon as some weights are even. A weighted complete intersection Xn(d1,…,dr;w)X_n(d_1,\dots,d_r;w) is the common zero locus of rr weighted-homogeneous polynomials of degrees d1,…,drd_1,\dots,d_r in P(w)\mathbb{P}(w). Which of these are genuinely smooth varieties is itself governed by arithmetic conditions on weights and degrees, and their Betti numbers and Euler characteristic are given by Dimca's (1985) monodromy computations. Two weaker regularity notions from the weighted literature often stand in for smoothness: a weighted complete intersection is quasismooth if its affine cone is smooth away from the cone point, and well-formed if it avoids the codimension-one singular strata of the ambient weighted projective space; both conditions are checked by divisibility arithmetic on the weights and degrees.

In this setting Wang and Wang (2012) proved the weighted analogue of the Kreck-type theorem: if Xn(d1,…,dr−1,dr;w)X_n(d_1,\dots,d_{r-1},d_r;w) and Xn(e1,…,er−1,dr;w)X_n(e_1,\dots,e_{r-1},d_r;w) are smooth weighted complete intersections of odd complex dimension n>2n>2, both contained in a common smooth weighted hypersurface Xn+r−1(dr;w)X_{n+r-1}(d_r;w), with weights ww pairwise relatively prime and odd, and νp(d/dr)≥2n+12(p−1)+1\nu_p(d/d_r)\geq\frac{2n+1}{2(p-1)}+1 for all primes pp with p(p−1)≤n+1p(p-1)\leq n+1 (where dd is the total degree), then the two manifolds are diffeomorphic if and only if they have the same total degree, Pontrjagin classes, and Euler characteristic. The proof compares the two manifolds as representatives of a class in a generalized bordism group encoded by a Thom spectrum (a spectrum whose homotopy groups form a bordism theory) and controls the mod-pp Adams filtration of the difference class — the descending filtration induced on homotopy or bordism groups by the Adams spectral sequence, the standard algebraic machine that computes such groups by successive approximation; the arithmetic hypothesis enters precisely at this filtration step. A companion paper (Wang and Wang 2010) classified the homotopy type of odd-dimensional smooth weighted complete intersections with pairwise coprime weights, but only when the total degree has no prime divisors smaller than n+32\frac{n+3}{2}. Every hypothesis of the 2012 theorem is load-bearing: even weights create 2-torsion in the ambient cohomology, non-coprime weights enlarge the ambient singular locus, even complex dimension introduces middle-dimensional intersection-form phenomena, and failure of the divisibility bound breaks the Adams filtration control.

Problem Statement

Let w=(ω0,…,ωn+r)w=(\omega_0,\dots,\omega_{n+r}) be positive integer weights and let Xn(d;w)⊂P(w)X_n(\mathbf d;w)\subset\mathbb{P}(w) be a smooth weighted complete intersection of complex dimension n≥3n\geq 3, with total degree dd the product of its multidegrees. Wang and Wang proved that two such intersections of odd dimension n>2n>2 sharing their last degree drd_r inside a common smooth weighted hypersurface Xn+r−1(dr;w)X_{n+r-1}(d_r;w) are diffeomorphic whenever they have the same total degree, Pontrjagin classes, and Euler characteristic, provided the weights are pairwise coprime and odd and νp(d/dr)≥2n+12(p−1)+1\nu_p(d/d_r)\geq\frac{2n+1}{2(p-1)}+1 for every prime pp with p(p−1)≤n+1p(p-1)\leq n+1. Determine whether the same Sullivan-type criterion — equality of total degree, Pontrjagin classes, and Euler characteristic implies diffeomorphism — extends to smooth weighted complete intersections beyond this regime, in any of the following directions: (a) even complex dimension n≥4n\geq 4; (b) weights that are not pairwise coprime, or not all odd; (c) multidegrees violating the stated mod-pp divisibility bound; (d) pairs not sharing a common smooth ambient weighted hypersurface of the stated form. The question is resolved either by proving the criterion for smooth weighted complete intersections in one of these regimes, or by an explicit pair of smooth weighted complete intersections with identical total degree, Pontrjagin classes, and Euler characteristic that is provably not diffeomorphic.

References

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