15 research outputs found
Embedding obstructions and 4-dimensional thickenings of 2-complexes
The vanishing of Van Kampen's obstruction is known to be necessary and
sufficient for embeddability of a simplicial n-complex into for , and it was recently shown to be incomplete for . We use
algebraic-topological invariants of four-manifolds with boundary to introduce a
sequence of higher embedding obstructions for a class of 2-complexes in .Comment: 10 pages; To appear in Proc. Amer. Math. So
Surgery on closed 4-manifolds with free fundamental group
Even though the disk embedding theorem is not available in dimension 4 for
free fundamental groups, some surgery problems may be shown to have topological
solutions. We prove that surgery problems may be solved if one considers closed
4-manifolds and the intersection pairing is extended from the integers, and
prove a related splitting result.Comment: 7 page
Exponential separation in 4-manifolds
We use a new geometric construction, grope splitting, to give a sharp bound
for separation of surfaces in 4-manifolds. We also describe applications of
this technique in link-homotopy theory, and to the problem of locating
pi_1-null surfaces in 4-manifolds. In our applications to link-homotopy, grope
splitting serves as a geometric substitute for the Milnor group.Comment: Published by Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol4/paper13.abs.htm
Alexander duality, gropes and link homotopy
We prove a geometric refinement of Alexander duality for certain 2-complexes,
the so-called gropes, embedded into 4-space. This refinement can be roughly
formulated as saying that 4-dimensional Alexander duality preserves the
disjoint Dwyer filtration. In addition, we give new proofs and extended
versions of two lemmas of Freedman and Lin which are of central importance in
the A-B-slice problem, the main open problem in the classification theory of
topological 4-manifolds. Our methods are group theoretical, rather than using
Massey products and Milnor \mu-invariants as in the original proofs.Comment: 19 pages. Published copy, also available at
http://www.maths.warwick.ac.uk/gt/GTVol1/paper5.abs.htm
Surgery and involutions on 4-manifolds
We prove that the canonical 4-dimensional surgery problems can be solved
after passing to a double cover. This contrasts the long-standing conjecture
about the validity of the topological surgery theorem for arbitrary fundamental
groups (without passing to a cover). As a corollary, the surgery conjecture is
reformulated in terms of the existence of free involutions on a certain class
of 4-manifolds. We consider this question and analyze its relation to the
A,B-slice problem.Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol5/agt-5-70.abs.htm
Subexponential groups in 4-manifold topology
We present a new, more elementary proof of the Freedman-Teichner result that
the geometric classification techniques (surgery, s-cobordism, and
pseudoisotopy) hold for topological 4-manifolds with groups of subexponential
growth. In an appendix Freedman and Teichner give a correction to their
original proof, and reformulate the growth estimates in terms of coarse
geometry.Comment: Published by Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol4/paper14.abs.htm