For each pair (m,n) of positive integers with (m,n)≠ (1,1) and an arbitrary field F with algebraic closure F, let Po^d,m_n( F) denote the space of m-tuples (f_1(z),⋯ ,f_m(z))∈ F [z]^m of F-coefficients monic polynomials of the same degree d such that the polynomials {f_k(z)}_k=1^m have no common root in F of multiplicity ≥ n. These spaces Po^d,m_n( F) were first defined and studied by B. Farb and J. Wolfson as generalizations of spaces first studied by Arnold, Vassiliev and Segal and others in several different contexts. In previous we determined explicitly the homotopy type of this space in the case F = C. In this paper, we investigate the case F = R.
In the first part of this work, we study Dirichlet-Voronoi domains for discrete isometry groups of Riemannian manifolds, in view of constructing cell structures on homogeneous (complete, real) flag manifolds, equivariant with respect to the action of the Weyl group. We give general results, allowing us to build such a structure from an admissible one on the domain. In particular, the injectivity radius plays a key role in the method. The second part starts with the computation of the injectivity radius of (real and complex) flag manifolds; a first step towards the application of the method developed in the first part. Then, with the help of the quaternion algebra, we investigate the particular case of the flag manifold O(3)/O(1)3 of SL3(R): we prove that the results of the first part apply and derive a new C53-equivariant cell structure on it, whose cellular complex of Z[C53]-modules is determined.
Let X and Y be closed G-manifolds and B C Y a closed invariant nonempty subset where G is a finite group. For any G-map f: X-* Y and for every subgroup H <= G, we introduce a Nielsen type number N (f H, BH) which is a lower bound for the number of connected components of WH-orbits of (f H )-1(BH). This theory generalizes existing Nielsen type numbers for various G and B with an application to the Nielsen Borsuk-Ulam theory for the minimal number of coincidences of f (x) = ft (x) where f : X-* Y and t a free involution on X.
The guts of a knot is an invariant defined for the knot complement by Agol-Zhang. Nearly fibered knots, which are defined as knots whose Floer homology has dimension two in the top Alexander grading, were introduced by Baldwin-Sivek. We provide three models for the guts of nearly fibered knots in the 3-sphere. As a corollary, the nearly fibered condition can be purely topologically characterized and is independent of the specific version of Floer theory.
We show that every closed orientable smooth 4-manifold admits a smooth embedding in a large class of closed 6-manifolds. In particular, we show that every smooth 4-manifold admits a smooth embedding in the complex projective 3-space. Our embedding technique also provides a new proof of embeddings of 4-manifolds in R7.
A slope p/q E Q is characterising for a knot K C S3 if the oriented homeomorphism type of the manifold S3 K(p/q) obtained by Dehn surgery of slope p/q on K uniquely determines the knot K. We combine analysis of JSJ decompositions with techniques involving lengths of shortest geodesics to find explicit conditions for a slope to be characterising for K in the case where K is any hyperbolic knot or any satellite knot by a hyperbolic pattern. Assuming that the list of 2-cusped orientable hyperbolic 3-manifolds obtained using the computer programme SnapPy is complete up to a certain point, we use hyperbolic volume inequalities to generate a refinement for the special case of Whitehead doubles. We also construct pairs of multiclasped Whitehead doubles of double twist knots for which 1/q is a noncharacterising slope.
We show that any product of bushy hyperbolic spaces has a unique coarse median structure, and that having a unique coarse median structure is a property closed under relative hyperbolicity. As a consequence, in contrast with the case of mapping class groups, there are non-hyperbolic pants graphs that have unique coarse median structures.
We show that the analog category of a finite group is essentially proportional to the size of its largest Sylow subgroup. We conclude that the universal upper bound given by the order of the group is very far from optimal.
We introduce a general definition of a n-crossed module of P-algebras over an algebraic operad P, which coincides with historical definitions in the cases of the operads As and Lie and n = 1. We establish a natural isomorphism between the abelian group of equivalence classes of n-crossed modules over a pair (A,M) for an operad P and the (n+1)^th operadic cohomology group of A with coefficients in M.
We study which lens spaces can bound smooth 4-manifolds with second Betti number one under various topological conditions. Specifically, we show that there are infinite families of lens spaces that bound compact, simply-connected, smooth 4-manifolds with second Betti number one, yet cannot bound a 4-manifold consisting of a single 0-handle and 2-handle. Additionally, we establish the existence of infinite families of lens spaces that bound compact, smooth 4-manifolds with first Betti number zero and second Betti number one, but cannot bound simply-connected 4-manifolds with second Betti number one. The construction of such 4-manifolds with lens space boundaries is motivated by the study of rational homology projective planes with cyclic quotient singularities.
This note provides the first example of a nontrivial connected component of the space of symplectic structures standard at infinity in dimension four.
Let G be a countable group acting properly on a metric space with contracting elements and {H-i :1 <= i <= n} be a finite collection of Morse subgroups in G. We prove that each Hi has infinite index in G if and only if the relative second bounded cohomology H-b(2) (G, {H-i}(n)(i=1); R) is infinite-dimensional. In addition, we also prove that for any contracting element g, there exists k > 0 such that H-b(2) (G, ((g(k)& Rang;; R) is infinite-dimensional. Our results generalize a theorem of Pagliantini-Rolli for finite-rank free groups and yield new results on the (relative) second bounded cohomology of groups.
We study the cohomology rings of tiling spaces Omega given by cubical substitutions. While there have been many calculations before of cohomology groups of such tiling spaces, the innovation here is that we use computer-assisted methods to compute the cup-product structure. This leads to examples of substitution tilings with isomorphic cohomology groups but different cohomology rings. Part of the interest in studying the cup product comes from Bellissard's gap-labeling conjecture, which is known to hold in dimensions <= 3, but where a proof is known in dimensions >= 4 only when the Chern character from K-0(Omega) to H*(Omega, Q) lands in H*(Omega, Z). Computation of the cup product on cohomology often makes it possible to compute the Chern character. We introduce a natural generalization of the gap-labeling conjecture, called the equivariant gap-labeling conjecture, which applies to tilings with a finite symmetry group. Again this holds in dimensions <= 3, but we are able to show that it fails in general in dimensions >= 4. This, plus some of our cup-product calculations, makes it plausible that the gap-labeling conjecture might fail in high dimensions.
We study properties of the cubical Joyal model structures on cubical sets by means of a combinatorial construction which allows for convenient comparisons between categories of cubical sets with and without symmetries. In particular, we prove that the cubical Joyal model structures on categories of cubical sets with connections are cartesian monoidal. Our techniques also allow us to prove that the geometric product of cubical sets (with or without connections) is symmetric up to natural weak equivalence in the cubical Joyal model structure, and to obtain induced model structures for (ac, 1)-categories on cubical sets with symmetries.
A knotted ribbon is one physical aspect of a knot. A folded ribbon knot is a depiction of a knot obtained by folding a long and thin rectangular strip to become flat. The ribbonlength of a knot type can be defined as the minimum length required to tie the given knot type as a folded ribbon knot. The ribbonlength has been conjectured to grow linearly or sublinearly with respect to a minimal crossing number. Several knot types provide evidence that this conjecture is true, but there is no proof for general cases. In this paper, we show that for any knot or link, the ribbonlength is bounded by a linear function of the crossing number. In more detail, Rib(K) <= 52 c(K) + 1 for a knot or link K. Our approach involves binary grid diagrams and bisected vertex leveling techniques.
We employ combinatorial techniques to present an explicit formula for the coefficients in front of Chern classes involving in the Hattori-Stong integrability conditions. We also give an evenness condition for the signature of stably almost-complex manifolds in terms of Chern numbers. As an application, it can be showed that the signature of a 2n-dimensional stably almost-complex manifold whose possibly nonzero Chern numbers being c_n and c_ic_n-i is even, which particularly rules out the existence of such structure on rational projective planes. Some other related results and remarks are also discussed in this article.
We obtain a sufficient condition for lattices in the automorphism group of a finite dimensional CAT(0) cube complex to have infinite girth. As a corollary, we get a version of Girth Alternative for groups acting geometrically: any such group is either {locally finite}-by-{virtually abelian} or it has infinite girth. We produce counterexamples to show that the alternative fails in the general class of groups acting cocompactly on finite dimensional CAT(0) cube complexes by obtaining examples of non virtually solvable groups which satisfy a law.
We realize every closed flat 3-manifold as a cusp section of a complete, finite-volume hyperbolic 4manifold whose symmetry group acts transitively on the set of cusps. Moreover, for every such 3-manifold, a dense subset of its flat metrics can be realized as cusp sections of a cusp-transitive 4-manifold. Finally, we prove that there are many 4-manifolds with pairwise isometric cusps for any given cusp type.
