40 research outputs found
Pulling Apart 2-spheres in 4-manifolds
An obstruction theory for representing homotopy classes of surfaces in
4-manifolds by immersions with pairwise disjoint images is developed, using the
theory of non-repeating Whitney towers. The accompanying higher-order
intersection invariants provide a geometric generalization of Milnor's
link-homotopy invariants, and can give the complete obstruction to pulling
apart 2-spheres in certain families of 4-manifolds. It is also shown that in an
arbitrary simply connected 4-manifold any number of parallel copies of an
immersed surface with vanishing self-intersection number can be pulled apart,
and that this is not always possible in the non-simply connected setting. The
order 1 intersection invariant is shown to be the complete obstruction to
pulling apart 2-spheres in any 4-manifold after taking connected sums with
finitely many copies of S^2\times S^2; and the order 2 intersection
indeterminacies for quadruples of immersed 2-spheres in a simply connected
4-manifold are shown to lead to interesting number theoretic questions.Comment: Revised to conform with the published version in Documenta
Mathematic
Homotopy versus isotopy: spheres with duals in 4-manifolds
David Gabai recently proved a smooth 4-dimensional "Light Bulb Theorem" in
the absence of 2-torsion in the fundamental group. We extend his result to
4-manifolds with arbitrary fundamental group by showing that an invariant of
Mike Freedman and Frank Quinn gives the complete obstruction to "homotopy
implies isotopy" for embedded 2-spheres which have a common geometric dual. The
invariant takes values in an Z/2Z-vector space generated by elements of order 2
in the fundamental group and has applications to unknotting numbers and
pseudo-isotopy classes of self-diffeomorphisms. Our methods also give an
alternative approach to Gabai's theorem using various maneuvers with Whitney
disks and a fundamental isotopy between surgeries along dual circles in an
orientable surface.Comment: Included into section 2 of this version is a proof that the operation
of `sliding a Whitney disk over itself' preserves the isotopy class of the
resulting Whitney move in the current setting. Some expository clarifications
have also been added. Main results and proofs are unchanged from the previous
version. 39 pages, 25 figure
The Group of Disjoint 2-Spheres in 4-Space
We compute the group of link homotopy classes of link maps of two 2-spheres
into 4-space. It turns out to be free abelian, generated by geometric
constructions applied to the Fenn-Rolfsen link map and detected by two
self-intersection invariants introduced by Paul Kirk in this setting. As a
corollary, we show that any link map with one topologically embedded component
is link homotopic to the unlink. Our proof introduces a new basic link
homotopy, which we call a Whitney homotopy, that shrinks an embedded Whitney
sphere constructed from four copies of a Whitney disk. Freedman's disk
embedding theorem is applied to get the necessary embedded Whitney disks, after
constructing sufficiently many accessory spheres as algebraic duals for
immersed Whitney disks. To construct these accessory spheres and immersed
Whitney disks we use the algebra of metabolic forms over the group ring Z[Z],
and introduce a number of new 4-dimensional constructions, including maneuvers
involving the boundary arcs of Whitney disks.Comment: This version significantly reorganized to account for referee's
report. Published version: Annals of Mathematics, November 2019 issue (Volume
190, no. 3) 76 pages, 54 figure
Milnor Invariants and Twisted Whitney Towers
This paper describes the relationship between the first non-vanishing Milnor
invariants of a classical link and the intersection invariant of a twisted
Whitney tower. This is a certain 2-complex in the 4-ball, built from immersed
disks bounded by the given link in the 3-sphere together with finitely many
`layers' of Whitney disks.
The intersection invariant is a higher-order generalization of the
intersection number between two immersed disks in the 4-ball, well known to
give the linking number of the link on the boundary, which measures
intersections among the Whitney disks and the disks bounding the given link,
together with information that measures the twists (framing obstructions) of
the Whitney disks.
This interpretation of Milnor invariants as higher-order intersection
invariants plays a key role in the classifications of both the framed and
twisted Whitney tower filtrations on link concordance (as sketched in this
paper). Here we show how to realize the higher-order Arf invariants, which also
play a role in the classifications, and derive new geometric characterizations
of links with vanishing Milnor invariants of length less than or equal to 2k.Comment: Typo corrected in statement of Theorem 16; no change to proof needed.
Otherwise, this revision conforms with the version published in the Journal
of Topology. 36 pages, 23 figure
Higher Order Intersections in Low-Dimensional Topology
We show how to measure the failure of the Whitney trick in dimension 4 by
constructing higher- order intersection invariants of Whitney towers built from
iterated Whitney disks on immersed surfaces in 4-manifolds. For Whitney towers
on immersed disks in the 4-ball, we identify some of these new invariants with
previously known link invariants like Milnor, Sato-Levine and Arf invariants.
We also define higher- order Sato-Levine and Arf invariants and show that these
invariants detect the obstructions to framing a twisted Whitney tower. Together
with Milnor invariants, these higher-order invariants are shown to classify the
existence of (twisted) Whitney towers of increasing order in the 4-ball. A
conjecture regarding the non- triviality of the higher-order Arf invariants is
formulated, and related implications for filtrations of string links and
3-dimensional homology cylinders are described. This article is an announcement
and summary of results to be published in several forthcoming papers