Equivalent non‐isotopic spheres in 4‐manifolds

We prove a concordance version of the 4‐dimensional light bulb theorem for π1‐negligible compact orientable surfaces, where there is a framed but not necessarily embedded dual sphere. That is, we show that if F0 and F1 are such surfaces in a 4‐manifold X that are homotopic and there exists an immersed framed 2‐sphere G in X intersecting F0 geometrically once, then F0 and F1 are concordant if and only if their Freedman–Quinn invariant prefixfq vanishes. The proof of the main result involves computing prefixfq in terms of intersections in the universal covering space and then applying work of Sunukjian in the simply‐connected case. This paper relies extensively on colour figures. Some references to colour may not be meaningful in the printed version, and we refer the reader to the online version which includes the colour figures.

“…In Theorem 3.1, the assumption that

π_{1} false(X false)

has no 2‐torsion is essential. Schwartz [10] later gave explicit examples of triples

(S_{0}, S_{1}, G)

of 2‐spheres in a 4‐manifold

X

with

π_{1} false(X false) ≅ Z / 2

that satisfy the other hypotheses of Theorem 3.1, but yet

S_{0}

and

S_{1}

are not isotopic. We discuss these examples in § 7.…”

Section: Four‐dimensional Motivationmentioning

confidence: 99%

“…Example 7.1 is due to Hannah Schwartz and it demonstrates the necessity of the condition that

f q = 0

. We use this example to illustrate our view of

f q

in terms of lifts of covers from [10].…”

Section: Examplesmentioning

confidence: 99%

“…We will first prove Theorem 1.4 in the case that

F_{0}

and

F_{1}

are 2‐spheres (in § 5) and then extend to positive‐genus surfaces in § 6. In § 7, we give three explicit examples showing: the necessity of

fq (F_{0}, F_{1}) = 0

(Example 7.1, due to Schwartz [10]); the necessity of the 2‐sphere

G

being framed (Example 7.2, constructed by the authors using Stong's [11]

prefixkm

invariant); the conclusion of concordance rather than isotopy (Example 7.3, due to Sato [8]). …”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Concordance of surfaces in 4‐manifolds and the Freedman–Quinn invariant

Klug

Miller

2021

“…Schwartz found infinitely many examples [9] which demonstrate that the assumption on π 1 (X) was essential. She produced 4-manifolds X with π 1 (X) ∼ = Z 2 , and pairs of homotopic embedded 2-spheres in X which share a transverse sphere but are not concordant.…”

Section: Introductionmentioning

confidence: 99%

“…Remark 5.2. (1) Schwartz [9] constructs pairs of embedded spheres in a 4-manifold X with 2-torsion in π 1 (X), which are regularly homotopic, and are such that any regular homotopy between them must contain a crossed Whitney move.…”

mentioning

confidence: 99%

Distances between surfaces in 4‐manifolds

Singh

2020

If normalΣ and Σ′ are homotopic embedded surfaces in a 4‐manifold, then they may be related by a regular homotopy (at the expense of introducing double points) or by a sequence of stabilisations and destabilisations (at the expense of adding genus). This naturally gives rise to two integer‐valued notions of distance between the embeddings: the singularity distance dsingfalse(normalΣ,Σ′false) and the stabilisation distance dprefixstfalse(normalΣ,Σ′false). Using techniques similar to those used by Gabai in his proof of the 4‐dimensional light bulb theorem, we prove that dprefixstfalse(normalΣ,Σ′false)⩽dsingfalse(normalΣ,Σ′false)+1.

Unknotting numbers of 2‐spheres in the 4‐sphere

Joseph

Klug

Benjamin

et al. 2021

Self Cite

We compare two naturally arising notions of ‘unknotting number’ for 2‐spheres in the 4‐sphere: namely, the minimal number of 1‐handle stabilizations needed to obtain an unknotted surface, and the minimal number of Whitney moves required in a regular homotopy to the unknotted 2‐sphere. We refer to these invariants as the stabilization number and the Casson–Whitney number of the sphere, respectively. Using both algebraic and geometric techniques, we show that the stabilization number is bounded above by one more than the Casson–Whitney number. We also provide explicit families of spheres for which these invariants are equal, as well as families for which they are distinct. Furthermore, we give additional bounds for both invariants, concrete examples of their non‐additivity, and applications to classical unknotting number of 1‐knots.