Journal of Topology最新文献

Birman–Menasco proved that there are finitely many knots having a given genus and braid index. We give a quantitative version of the Birman–Menasco finiteness theorem, an estimate of the crossing number of knots in terms of genus and braid index. As applications, we give a solution of the braid index problem, the problem to determine the braid index of a given link, and provide estimates of the crossing number of connected sums or satellites.

We compute some $�R$-motivic stable homotopy groups. For $��s�-�w�⩽�11�$, we describe the motivic stable homotopy groups $�{\pi }_{�s�,�w�}$ of a completion of the $�R$-motivic sphere spectrum. We apply the $�\rho$-Bockstein spectral sequence to obtain $�R$-motivic $�Ext$ groups from the $�C$-motivic $�Ext$ groups, which are well understood in a large range. These $�Ext$ groups are the input to the $�R$-motivic Adams spectral sequence. We fully analyze the Adams differentials in a range, and we also analyze hidden extensions by $�\rho$, 2, and $�\eta$. As a consequence of our computations, we recover Mahowald invariants of many low-dimensional classical stable homotopy elements.

Bivariant theory is a unified framework for cohomology and Borel–Moore homology theories. In this paper, we extract an $�\infty$-enhanced bivariant homology theory from Gaitsgory–Rozenblyum's six functor formalism.

We study the equivariant genera of strongly invertible and periodic knots. Our techniques include some new strongly invertible concordance group invariants, Donaldson's theorem, and the g-signature. We find many new examples where the equivariant 4-genus is larger than the 4-genus.

For any oriented cusped hyperbolic 3-manifold $�M$, we study its $��(�R�,�\epsilon �)�$-panted cobordism group, which is the abelian group generated by $��(�R�,�\epsilon �)�$-good curves in $�M$ modulo the oriented boundaries of $��(�R�,�\epsilon �)�$-good pants. In particular, we prove that for sufficiently small $��\epsilon �>�0�$ and sufficiently large $��R�>�0�$, some modified version of the $��(�R�,�\epsilon �)�$-panted cobordism group of $�M$ is isomorphic to $��H_1��(�\text{SO}��$

Let $��M�=�W�{\cup }_T�V�$ be an amalgamation of two compact 3-manifolds along a torus, where $�W$ is the exterior of a knot in a homology sphere. Let $�N$ be the manifold obtained by replacing $�W$ with a solid torus such that the boundary of a Seifert surface in $�W$ is a meridian of the solid torus. This means that there is a degree-one map $��f�:�M�\to �N�$, pinching $�W$ into a solid torus while fixing $�V$. We prove that $��g�\left(�M�\right)�⩾�g�\left(�N�\right)�$, where $��g�\left(�M�\right)�$ denotes the Heegaard genus. An immediate corollary is that the tunnel number of a satellite knot is at least as large as the tunnel number of its pattern knot.

In a previous paper, we showed that the original definition of cylindrical contact homology, with rational coefficients, is valid on a closed three-manifold with a dynamically convex contact form. However, we did not show that this cylindrical contact homology is an invariant of the contact structure. In the present paper, we define ‘nonequivariant contact homology’ and ‘$�S^1$-equivariant contact homology’, both with integer coefficients, for a contact form on a closed manifold in any dimension with no contractible Reeb orbits. We prove that these contact homologies depend only on the contact structure. Our construction uses Morse–Bott theory and is related to the positive $�S^1$-equivariant symplectic homology of Bourgeois-Oancea. However, instead of working with Hamiltonian Floer homology, we work directly in contact geometry, using families of almost complex structures. When cylindrical contact homology can also be defined, it agrees with the tensor product of the $�S^1$-equivariant contact homology with $�Q$. We also present examples showing that the $�S^1$-equivariant contact homology contains interesting torsion information. In a subsequent paper, we will use obstruction bundle gluing to extend the above story to closed three-manifolds with dynamically convex contact forms, which in particular will prove that their cylindrical contact homology has a lift to integer coefficients which depends only on the contact structure.

We introduce a global equivariant refinement of algebraic K-theory; here ‘global equivariant’ refers to simultaneous and compatible actions of all finite groups. Our construction turns a specific kind of categorical input data into a global $�\Omega$-spectrum that keeps track of genuine $�G$-equivariant infinite loop spaces, for all finite groups $�G$. The resulting global algebraic K-theory spectrum is a rigid way of packaging the representation K-theory, or ‘Swan K-theory’ into one highly structured object.

We give a short and elementary proof of the nonrealizability of the mapping class group via homeomorphisms. This was originally established by Markovic, resolving a conjecture of Thurston. With the tools established in this paper, we also obtain some rigidity results for actions of the mapping class group on Euclidean spaces.

For a half-translation surface $��(�S�,�q�)�$, the associated saddle connection complex $��A�(�S�,�q�)�$ is the simplicial complex where vertices are the saddle connections on $��(�S�,�q�)�$, with simplices spanned by sets of pairwise disjoint saddle connections. This complex can be naturally regarded as an induced subcomplex of the arc complex. We prove that any simplicial isomorphism $��\varphi �:�A��(�S�,�q�)��\to �A��(�S^{\prime }�,�q^{\prime }�)��$ between saddle connection complexes is induced by an affine diffeomorphism $��F�:��(�S�,�q�)��\to ��(�S^{\prime }�,�q^{}$