We exhibit an explicit delooping of the space of string links with components in the hypercube , for and .
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The homotopy fiber of the inclusion from the long embedding space to the long immersion space is known to be an iterated based loop space (if the codimension is greater than two). In this paper we deloop the homotopy fiber to obtain the topological Stiefel manifold, combining results of Lashof and of Lees. We also give…
Three applications of delooping to h-principles on manifolds.
problem Proving h-principles on closed manifolds under different conditions.
method A method to prove h-principles on closed manifolds under varying conditions.
result Homotopical version of Vassiliev's h-principle, contractibility of framed functions space, and Mather-Thurston theory version.
Dehn twists around simple closed curves in oriented surfaces satisfy the braid relations. This gives rise to a group theoretic from the braid group to the mapping class group. We prove here that this map is trivial in stable homology with any trivial coefficients. In particular this proves an old conjecture of J. Harer…
The smoothing theory is revised to generalize to different disc embedding spaces.
problem Generalizing the smoothing theory to various disc embedding spaces.
method Revising the Morlet-Burghelea-Lashof-Kirby-Siebenmann theorem to apply to different versions of disc smooth embedding spaces.
result The delooping of disc embedding spaces is shown to be compatible with the Hatcher and Budney actions.
New proof of Genauer fibration sequence in cobordism categories.
problem Relating cobordism categories of closed and boundary-manifolds.
method Inspired from algebraic K-theory, using cobordism categories and delooping.
result Generalization of Waldhausen's Additivity theorem to cobordism categories.
We construct a model of differential K-theory, using the geometrically defined Chern forms, whose cocycles are certain equivalence classes of maps into the Grassmannians and unitary groups. In particular, we produce the circle-integration maps for these models using classical homotopy-theoretic constructions, by incorp…
Tillmann introduced two infinite loop space structures on the plus construction of the classifying space of the stable mapping class group, each with different computational advantages. The first one uses disjoint union on a suitable cobordism category, whereas the second uses an operad which extends the pair of pants …
The study proves spaces of positive scalar curvature metrics have infinite loop space homotopy type.
problem Understanding spaces of positive scalar curvature metrics and their homotopy properties.
method Cobordism category analysis, Quillen's Theorem B, Gromov-Lawson surgery theorem, Segal's theory of Γ-spaces.
result Spaces of positive scalar curvature metrics have the homotopy type of infinite loop spaces.
Proves Khovanov homology conjecture for 3-stranded braids.
problem Khovanov homology of 3-stranded braid closures.
method Bar-Natan's version of Khovanov homology for tangles, delooping, cancellation techniques, and spectral sequences.
result Khovanov homology of 3-stranded braid closures contains only 2-torsion.
In this paper, we give a quantum interpretation of the Bismut-Chern character form (the loop space lifting of the Chern character form) as well as the Chern character form associated to a complex vector bundle with connection over a smooth manifold in the framework of supersymmetric quantum field theories developed by …
New approach connects pseudoisotopy theory to algebraic K-theory.
problem Understanding pseudoisotopy theory using algebraic K-theory.
method Constructing a zig-zag from pseudoisotopy space to algebraic K-theory space, using homological computations.
result Maps are p-locally (2n-4)-connected for large primes p.
The paper simplifies embedding spaces in manifolds by attaching handles.
problem Embedding spaces in manifolds with knotted spheres.
method Using framed dual spheres and attaching handles to simplify embedding spaces.
result The homotopy type of embedding spaces is significantly simplified.
Study of embedding calculus using infinite operads.
problem Spaces of embeddings and automorphisms of manifolds.
method Infinite operadic towers and Morita -categories.
result Generalization of embedding calculus to bordism categories.
Study the moduli space of reducible 3-manifolds using prime decomposition.
problem Understanding the homotopy type of moduli spaces of reducible 3-manifolds.
method Construct a splitting map from to , yielding a prime decomposition fibre sequence.
result The fibre is a finite, connected cell complex, and the prime decomposition fibre sequence is effective for computations.