We introduce the concept of a bridge trisection of a neatly embedded surface in a compact four-manifold, generalizing previous work with Alexander Zupan in the setting of closed surfaces in closed four-manifolds. Our main result states that any neatly embedded surface Ᏺ in a compact four-manifold X can be isotoped to lie in bridge trisected position with respect to any trisection T of X. A bridge trisection of Ᏺ induces a braiding of the link @Ᏺ with respect to the open-book decomposition of @X induced by T , and we show that the bridge trisection of Ᏺ can be assumed to induce any such braiding. We work in the general setting in which @X may be disconnected, and we describe how to encode bridge trisected surface diagrammatically using shadow diagrams. We use shadow diagrams to show how bridge trisected surfaces can be glued along portions of their boundary, and we explain how the data of the braiding of the boundary link can be recovered from a shadow diagram. Throughout, numerous examples and illustrations are given. We give a set of moves that we conjecture suffice to relate any two shadow diagrams corresponding to a given surface. We devote extra attention to the setting of surfaces in B 4 , where we give an independent proof of the existence of bridge trisections and develop a second diagrammatic approach using tri-plane diagrams. We characterize bridge trisections of ribbon surfaces in terms of their complexity parameters. The process of passing between bridge trisections and band presentations for surfaces in B 4 is addressed in detail and presented with many examples. 57K10, 57K40, 57K45

We investigate cohomological properties of fundamental groups of strictly developable simple complexes of groups X. We obtain a polyhedral complex equivariantly homotopy equivalent to X of the lowest possible dimension. As applications, we obtain a simple formula for proper cohomological dimension of CAT.0/ groups whose actions admit a strict fundamental domain; for any building of type (W,S) that admits a chamber transitive action by a discrete group, we give a realisation of the building of the lowest possible dimension equal to the virtual cohomological dimension of W; under general assumptions, we confirm a folklore conjecture on the equality of Bredon geometric and cohomological dimensions in dimension one; finally, we give a new family of counterexamples to the strong form of Brown’s conjecture on the equality of virtual cohomological dimension and Bredon cohomological dimension for proper actions.

M / be the space of smooth embeddings from the circle to a closed manifold M . We introduce a new spectral sequence converging to H .Emb.S 1 ; M // for a simply connected closed manifold M of dimension 4 or more, which has an explicit E 1 -page and a computable E 2 -page. As applications, we compute some part of the cohomology for M D S k S l with some conditions on the dimensions k and l, and prove that the inclusion Emb.S 1 ; M / ! Imm.S 1 ; M / to the immersions induces an isomorphism on 1 for some simply connected 4-manifolds. This gives a restriction on a question posed by Arone and Szymik. The idea to construct the spectral sequence is to combine a version of Sinha's cosimplicial model for the knot space and a spectral sequence for a configuration space by Bendersky and Gitler. The cosimplicial model consists of configuration spaces of points (with a tangent vector) in M . We use Atiyah duality to transfer the structure maps on the configuration spaces to maps on Thom spectra of the quotient of a direct product of M by the fat diagonal. This transferred structure is the key to defining our spectral sequence, and is also used to show that Sinha's model can be resolved into simpler pieces in a stable category.

The category of complete differential graded Lie algebras provides nice algebraic models for the rational homotopy types of nonsimply connected spaces. In particular, there is a realization functor, h i, of any complete differential graded Lie algebra as a simplicial set. In a previous article, we considered the particular case of a complete graded Lie algebra, L 0 , concentrated in degree 0 and proved that hL 0 i is isomorphic to the usual bar construction on the Maltsev group associated to L 0 . Here we consider the case of a complete differential graded Lie algebra, L D L 0 ˚L1 , concentrated in degrees 0 and 1. We establish that the category of such two-stage Lie algebras is equivalent to explicit subcategories of crossed modules and Lie algebra crossed modules, extending the equivalence between pronilpotent Lie algebras and Maltsev groups. In particular, there is a crossed module Ꮿ.L/ associated to L. We prove that Ꮿ.L/ is isomorphic to the Whitehead crossed module associated to the simplicial pair .hLi; hL 0 i/. Our main result is the identification of hLi with the classifying space of Ꮿ.L/.

Knot Floer homology and link Floer homology are invariants of links in S 3 , defined using a version of Lagrangian Floer homology [31; 32; 34]. They are categorifications of the single variable and multivariable Alexander polynomials, respectively. Here we briefly highlight the key features of knot Floer homology and link Floer homology that we use to obtain our detection results. We work with coefficients in Z=2Z.

We construct a well-defined relative second grading on symplectic Khovanov cohomology from holomorphic disc counting. We show that it recovers the Jones grading of Khovanov homology up to an overall grading shift over any characteristic zero field, through proving that the isomorphism of Abouzaid-Smith can be refined as an isomorphism between bigraded cohomology theories. We prove it by constructing an exact triangle of symplectic Khovanov cohomology that behaves similarly to the unoriented skein exact triangle for Khovanov homology. We use a version of symplectic Khovanov cohomology defined for bridge diagrams and obtain an absolute homological grading in this construction.

We prove a homological stability theorem for the subgroup of the mapping class group acting as the identity on some fixed portion of the first homology group of the surface. We also prove a similar theorem for the subgroup of the mapping class group preserving a fixed map from the fundamental group to a finite group, which can be viewed as a mapping class group version of a theorem of Ellenberg-Venkatesh-Westerland about braid groups. These results require studying various simplicial complexes formed by subsurfaces of the surface, generalizing work of Hatcher-Vogtmann.

We introduce and study a new family of extensions for the Borsuk-Ulam and\ntopological Radon type theorems. The defining idea for this new family is to\nreplace requirements of the form `a subset that is large in some sense goes to\na singleton' with requirements of the milder form `a subset that is large in\nsome sense goes to a subset that is small in some sense'. This approach covers\nthe case of mappings m-sphere to n-space with m<n and extends to wider classes\nof spaces.\n

The set of isotopy classes of ordered n-component links in the 3-sphere is\nacted on by the symmetric group via permutation of the components. The\nintrinsic symmetry group of the link, S(L), is defined to be the set of\nelements in the symmetric group that preserve the ordered isotopy type of L as\nan unoriented link. The study of these groups was initiated in 1969, but the\nquestion of whether or not every subgroup of the symmetric group arises as an\nintrinisic symmetry group of some link has remained open. We provide\ncounterexamples; in particular, if n > 5, then there does not exist an\nn-component link L for which S(L) is the alternating group.\n