1,720,981 research outputs found
Counterexamples in 4-manifold topology
Full text linkWe illustrate the rich landscape of 4-manifold topology through the lens of
counterexamples. We consider several of the most commonly studied equivalence
relations on 4-manifolds and how they are related to one another. We explain
implications e.g. that -cobordant manifolds are stably homeomorphic, and we
provide examples illustrating the failure of other potential implications. The
information is conveniently organised in a flowchart and a table.Comment: 37 pages, 4 figures, 1 table; in v2, we have made several changes in
response to a referee report, including writing a more detailed introduction,
adding more details about the surgery exact sequence, uniformising the
structure of the subsections describing counterexamples, and adding
Proposition 5.6. This is the version published in EMS Survey
A family of non-split topologically slice links with arbitrarily large smooth slice genus
No full textWe construct an infinite family of topologically slice 2-component boundary links l(i), none of which is smoothly concordant to a split link, such that g(4)(l(i)) = i.
Embedding surfaces in 4-manifolds
Full text linkWe prove a surface embedding theorem for 4-manifolds with good fundamental group in the presence of dual spheres, with no restriction on the normal bundles. The new obstruction is a Kervaire-Milnor invariant for surfaces and we give a combinatorial formula for its computation. For this we introduce the notion of band characteristic surfaces.56 pages, 20 figures; in v2, we have added a new section (Section 1.5) containing applications to knot theory; v3: several minor changes and corrections following a referee report, to appear in Geometry & Topolog
On the Upsilon invariant and satellite knots
Full text linkWe study the effect of satellite operations on the Upsilon invariant of Ozsvath-Stipsicz-Szabo. We obtain results concerning when a knot and its satellites are independent; for example, we show that the set is a basis for an infinite rank summand of the group of smooth concordance classes of topologically slice knots, for D the positive clasped untwisted Whitehead double of any knot with positivetau-invariant, e.g. the right-handed trefoil. We also prove that the image of the Mazur satellite operator on the smooth knot concordance group contains an infinite rank subgroup of topologically slice knots.<br
The Collar Adding Lemma
No full textThe collar adding lemma is a key ingredient in the proof of the disc embedding theorem. Specifically, it proves that a skyscraper with an added collar is homeomorphic to the standard 4-dimensional 2-handle. The proof is similar to the proof in a previous chapter that the Alexander gored ball with an added collar is homeomorphic to the standard 3-ball. Roughly speaking, a skyscraper is seen as the quotient space of the 4-ball corresponding to a certain decomposition. The added collar allows the decomposition to be modified so that the resulting decomposition shrinks; that is, the corresponding quotient space, which is identified with the skyscraper with an added collar, is homeomorphic to the original 4-ball
Concordance of knots in S1×S2
Full text linkWe establish a number of results about smooth and topological concordance of knots in S1xS2. The winding number of a knot in S1xS2 is defined to be its class in H1(S1xS2;Z)Z. We show that there is a unique smooth concordance class of knots with winding number one. This improves the corresponding result of Friedl-Nagel-Orson-Powell in the topological category. We say a knot in S1xS2 is slice (respectively, topologically slice) if it bounds a smooth (respectively, locally flat) disk in D2xS2. We show that there are infinitely many topological concordance classes of non-slice knots, and moreover, for any winding number other than +/- 1, there are infinitely many topological concordance classes even within the collection of slice knots. Additionally, we demonstrate the distinction between the smooth and topological categories by constructing infinite families of slice knots that are pairwise topologically but not smoothly concordant, as well as non-slice knots that are topologically slice and are pairwise topologically, but not smoothly, concordant.
Skyscrapers are standard: the details
No full text‘Skyscrapers Are Standard: The Details’ provides a thorough and detailed proof that every skyscraper is homeomorphic to the standard 2-handle, relative to the attaching region. Results from decomposition space theory established in Part I and the constructive results from Part II are combined. The idea is to construct a subset of a skyscraper called the design, define an embedding of this subset into the standard 2-handle, and then consider the decomposition spaces obtained by quotienting out the connected components of the complement of this common subset. It is shown that the decomposition spaces are homeomorphic, and that both quotient maps are approximable by homeomorphisms. This chapter also shows that everything can be done fixing a neighbourhood of the attaching region. It is then deduced that skyscrapers are standard, as desired
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
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