Geometry & Topology
18 results · 26 papers · 3 with a Lean formalization
Every locally compact second-countable Hausdorff group acting faithfully and jointly continuously on a connected finite-dimensional topological manifold is a Lie group. This proves the Hilbert–Smith conjecture in all finite dimensions, for Hausdorff second-countable manifolds without boundary.
1 paperThe unrestricted four-dimensional disk-embedding conjecture fails: framed algebraic dual spheres do not suffice to obtain disjoint locally flat spanning disks. In particular, the free group F2 is not good in the sense of Freedman–Quinn. Also constructs a finitely presented integral Poincaré duality group of dimension four with a finite classifying space but no realization as the fundamental group of a closed aspherical topological four-manifold, disproving Wall's conjecture.
3 papersDistinct Dehn surgery slopes on a nontrivial smooth knot in S3 never produce orientation-preservingly homeomorphic manifolds, proving the purely cosmetic surgery conjecture. The statement includes the meridional slope.
1 paperConstructs a uniformly discrete bounded-geometry space whose maximal coarse assembly map is not rationally injective. The example is a coarse disjoint union of finite connected graphs of uniformly bounded degree, with an infinite-order kernel class. A companion gives the analogous failure for reduced coarse assembly, disproving the rational coarse Novikov conjecture.
2 papersLeanFor every n ≥ 0, constructs a finite Smith–Toda spectrum at a prime p depending on n, with Brown–Peterson homology \(BP_*/(p,v_1,\ldots,v_n)\) and the canonical comodule structure. Thus Smith–Toda complexes exist at every height when the prime may vary; an explicit example realizes \(V(4)\) at p = 1009.
2 papersResolves the odd-primary Kervaire invariant problem at the prime three: exactly the standard classes with indices 0, 2, and 3 survive in the mod-three Adams spectral sequence, in stems 10, 106, and 322. Each surviving detection coset contains an element of exact additive order three.
1 paperProves the rational-homology form of Quillen's conjecture for every finite group and every prime. If the largest normal p-subgroup of G is trivial, the poset of nontrivial elementary abelian p-subgroups has nonzero augmented reduced rational homology and is therefore not contractible.
1 paperProves Chai's invariant-ideal conjecture for Lubin–Tate deformation rings over finite residue fields, at every prime and positive height n. Through the implication of Barthel–Heard–Naumann, this proves the Hovey–Strickland conjecture: dualizable \(K(n)\)-local spectra have exactly \(n+2\) thick tensor ideals, and their Balmer spectrum is a chain of \(n+1\) points.
1 paperProves the Grothendieck homotopy hypothesis for ∞-groupoids associated with every Grothendieck coherator in the Ara–Henry convention: these algebraic objects recover the homotopy theory of spaces.
1 paperLeanAnswers the degreewise finiteness question of Hovey and Hovey–Strickland: every homotopy group of the \(K(n)\)-local sphere is a finitely generated ℤp-module, for every prime, positive height and integer degree. The same conclusion holds after \(K(n)\)-localizing any finite p-local spectrum.
1 paperDetermines the optimal chromatic loss from geometric H-fixed points to geometric G-fixed points for every subgroup H of a finite p-group G. At every nonnegative height, the loss equals the shortest subnormal-chain length from H to G with cyclic quotients. Each quotient counts once regardless of order, and finite spectra witness sharpness.
1 paperProves that the L2-Betti numbers of the universal cover of every closed connected aspherical topological four-manifold vanish outside degree two. More generally, the same conclusion holds for every finite connected aspherical integral Poincaré complex of formal dimension four, proving the four-dimensional Singer conjecture in this wider class.
1 paperProves Curtis’s conjecture: the positive-degree mod-two stable Hurewicz image of the sphere is spanned by the images of the Hopf-invariant-one classes η, ν, σ and the Kervaire-invariant-one classes that exist.
1 paperResolves the Ara–Maltsiniotis conjecture: for every n ≥ 1 and n = ω, small strict globular n-categories admit proper combinatorial Thomason model structures Quillen equivalent to simplicial sets. Thus strict higher categories model the homotopy theory of spaces in every stated dimension.
1 paperLeanDisproves strong chromatic splitting at height three for primes p ≥ 5, and weak splitting for the derived p-completed sphere at heights p (p ≥ 5) and \(p+1\) (p ≥ 7). Nevertheless, for n ≥ 1 and \(p\gt n+1\), the overlap \(L_{n-1}L_{K(n)}S_p^\wedge\) admits a \(2^n\)-stage filtration by the predicted localized-sphere pieces. At height three and prime three, even finite assembly from such pieces fails in the category of \(E(2)\)-local modules over the derived completed sphere.
5 papersRefutes the Hahn–Wilson conjecture at chromatic height two. For every sufficiently large prime p, constructs a connective p-complete spectrum of exact fp-type two that cannot be built from completed \(\mathrm{BP}\langle2\rangle\) by finitely many sums, shifts, cones and retracts. The examples nevertheless satisfy the finite and telescopic localization comparisons.
1 paperConstructs closed connected aspherical topological four-manifolds that are homotopy equivalent but not homeomorphic, with a common word-hyperbolic fundamental group. This disproves even the homeomorphism-existence formulation of the Borel conjecture in dimension four.
1 paperConstructs a finite connected three-dimensional CW complex whose universal cover has no integral homology above degree two and whose third cohomology vanishes for every local coefficient module, but which has no finite two-dimensional homotopy model. This disproves Wall's finite D(2) conjecture; the example has infinite fundamental group.
1 paperNo result matches this filter yet.