Topology

We prove the Hilbert–Smith conjecture in every finite dimension: every locally compact second-countable Hausdorff group acting faithfully and jointly continuously on a connected Hausdorff second-countable finite-dimensional topological manifold without boundary is a Lie group.

We disprove the unrestricted four-dimensional disk-embedding conjecture. We construct immersed disks in a compact oriented smooth four-manifold with framed algebraic dual spheres satisfying the usual equivariant intersection and reduced self-intersection conditions, but with no pairwise disjoint locally flat replacements that preserve the boundary maps and induced normal framings. The obstruction holds even when the replacement disks' relative homotopy classes are not prescribed.

We disprove the four-dimensional disc embedding conjecture without a fundamental-group hypothesis, even when no homotopy classes or output framings are prescribed. We construct a compact oriented smooth four-manifold containing finitely many disc maps with framed algebraic dual spheres whose boundary circles bound no disjoint locally flat discs. Consequently, the free group on two generators is not good in the sense of Freedman–Quinn, and neither is any group containing it as a subgroup.

We construct a finitely presented integral Poincaré duality group of dimension four that has a finite classifying space but is not the fundamental group of any closed aspherical topological four-manifold. This gives a negative answer to Wall's manifold-realization question in dimension four.

We prove the purely cosmetic surgery conjecture for knots in the three-sphere. Distinct Dehn surgery slopes on a nontrivial smooth knot never give orientation-preservingly homeomorphic manifolds.

We construct a uniformly discrete bounded-geometry space whose maximal coarse assembly map has an infinite-order element in its kernel. The space is a coarse disjoint union of finite connected graphs of uniformly bounded degree, so maximal coarse assembly need not be rationally injective even for such graph unions.

We disprove the coarse Novikov conjecture: ordinary coarse assembly need not be rationally injective for uniformly discrete spaces of bounded geometry. We construct a coarse disjoint union of finite graphs of uniformly bounded degree and an infinite-order class in its degree-one coarse K-homology whose image under ordinary coarse assembly in the K-theory of the reduced, locally compact Roe algebra vanishes.

For every nonnegative integer n, we construct a Smith–Toda complex \(V(n)\) at some prime p depending on n. It is a finite p-local spectrum whose Brown–Peterson homology is \(\mathrm{BP}_*/(p,v_1,\ldots,v_n)\) with its canonical comodule structure and generator in degree zero. Each listed generator is killed to its first power.

We construct a Smith–Toda complex \(V(4)\) at the prime 1009. It is an ordinary finite 1009-local spectrum whose Brown–Peterson homology is \(BP_*/(1009,v_1,v_2,v_3,v_4)\), with its canonical comodule structure and generator in degree zero.

We solve the Kervaire invariant problem at the prime three for the standard Kervaire classes in the mod-three Adams spectral sequence. These classes survive precisely at indices 0, 2, and 3, and each surviving detection coset contains an element of additive order three. In particular, there is an order-three Kervaire element in stem 322.

We prove Quillen's conjecture for all finite groups and all primes. More precisely, if a finite group G has trivial largest normal p-subgroup \(O_p(G)\), then the poset of nontrivial elementary abelian p-subgroups of G has nonzero augmented reduced rational homology. This establishes the stronger rational-homology form of the conjecture.

We classify the prime and radical ideals of the Lubin–Tate deformation ring invariant under an open Morava stabilizer subgroup, at every prime and positive height. This proves Chai's invariant-ideal conjecture in the standard finite-residue-field formulation. It also proves the Hovey–Strickland conjecture: the dualizable \(K(n)\)-local category has exactly \(n+2\) thick tensor ideals, and its Balmer spectrum is a chain of \(n+1\) points.

We prove the Grothendieck homotopy hypothesis for every Grothendieck coherator in the Ara–Henry convention: its weak globular infinity-groupoids recover the homotopy theory of spaces. We also resolve Henry's pushout conjecture, showing that elementary expansions preserve components and all homotopy groups of cellular infinity-groupoids.

We prove that the homotopy groups of the \(K(n)\)-local sphere are finitely generated ℤ_p-modules in every integer degree, for every prime p and positive height n. Equivalently, the same conclusion holds after \(K(n)\)-localizing any finite p-local spectrum. This answers the degreewise finiteness question of Hovey and Hovey–Strickland.

For a finite p-group G and a subgroup H, we prove that the optimal chromatic fixed-point loss equals the shortest length of a subnormal chain from H to G with cyclic quotients. The equality holds at every prime and every nonnegative height, resolving positively the equality proposed by Kuhn and Lloyd.

We prove the four-dimensional Singer conjecture: the L²-Betti numbers of the universal cover of a closed connected aspherical topological four-manifold vanish outside degree two. More generally, the same vanishing holds for finite connected aspherical integral Poincaré complexes of formal dimension four, including nonorientable ones.

We prove Curtis's conjecture: in every positive degree, the mod-two stable Hurewicz image of the sphere is spanned by the images of η, ν, σ and the Kervaire-invariant-one classes that exist. Consequently, Eccles's conjecture holds for every sphere Sⁿ with n > 0.

We prove the higher-dimensional Thomason model-structure conjecture of Ara and Maltsiniotis. For every \(1\le n\le\infty\), the category of small strict globular n-categories admits a proper combinatorial model structure that is Quillen equivalent to simplicial sets. Its weak equivalences and fibrations are detected by the twice-extended Street nerve \(\mathrm{Ex}^2N_n\), and the Quillen equivalence is given by \(c_n\mathrm{Sd}^2\dashv\mathrm{Ex}^2N_n\). Thus strict higher categories model the homotopy theory of spaces in every positive finite dimension and in dimension ω.

For every prime p ≥ 5, we construct an explicit eight-stage filtration of \(L_2L_{K(3)}\mathbb S_p^\wedge\) by the local-sphere layers in the height-three chromatic-splitting pattern. The map from its first stage to the overlap is the canonical unit, and two signed fracture formulas identify all attachments for the chosen local maps and compatibility homotopy. A companion canonical-map theorem further implies that the first height-one attachment is nonzero.

For the derived p-completion of the sphere, the canonical weak chromatic splitting map has no homotopy retraction at height p for every prime p ≥ 5, and at height \(p+1\) for every prime p ≥ 7. The classical sphere product \(\beta_1^{(p-1)^2}\) gives a nonzero kernel class in both ranges.

For every n ≥ 1 and prime \(p\gt n+1\), we construct a \(2^n\)-stage ordered filtration of \(L_{n-1}L_{K(n)}S_p^\wedge\) with the classical chromatic-splitting cofibers. The map from the first stage to the target is the canonical localization unit.

At the prime three and height three, the chromatic overlap cannot be constructed from the rational, height-one, and height-two local spheres by finitely many sums, shifts, cofibers, and retracts in the category of \(E(2)\)-local modules over the derived 3-complete sphere. This gives a negative answer to the ordinary finite-assembly question for chromatic overlaps.

For every prime p ≥ 5, the canonical map \(L_0L_{K(3)}S\to L_0L_{K(2)}L_{K(3)}S\) for the sphere spectrum S is nonzero on π₋₃. Consequently, the height-three strong chromatic splitting formula is false in this range, even as an equivalence of underlying \(E(2)\)-local spectra without specified summand maps.

We disprove the Hahn–Wilson conjecture at height two. For every sufficiently large prime p, we construct a connective p-complete spectrum X of exact fp-type two outside the ordinary thick subcategory generated by the specified standard form of \(\mathrm{BP}\langle2\rangle_p^\wedge\). The same spectrum satisfies both localization comparisons \(L_2^fX\simeq L_2X\) and \(L_{T(2)}X\simeq L_{K(2)}X\).

We construct closed connected aspherical topological four-manifolds that are homotopy equivalent but not homeomorphic, disproving the homeomorphism-existence formulation of the Borel conjecture in dimension four. Their common fundamental group is word-hyperbolic. One of the manifolds also has a self-homotopy equivalence not homotopic to a homeomorphism.

We give a negative answer to Wall's finite \(D(2)\) problem. We construct a finite connected three-dimensional CW complex satisfying the \(D(2)\) finiteness condition but not homotopy equivalent to any finite CW complex of dimension at most two. The example has infinite fundamental group.