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Engel relations in 4-manifold topology

Slava Krushkal

August 29, 2015

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I. Group theoryLet G be a group normally generated by a fixed finite collection ofelements g1, . . . , gn.

The Milnor group of G, defined with respect to a given normalgenerating set {gi}, is given by

MG := G/ ⟨⟨ [gi, gyi ] i = 1, . . . , n, y ∈ G⟩⟩. (1)

(Each gi commutes with all of its conjugates gyi .)

Main example:

L is an n-component link in S3, G = π1(S3 r L).

Normal generators gi: meridians to the components li of L.

Then MG, defined with respect to the {gi}, is called the Milnorgroup ML of the link L.

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Two links are link-homotopic if they are connected by a1-parameter family of link maps where different components staydisjoint for all values of the parameter.

Figure: A non-generic “crossing-time” during a link homotopy.

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MG = G/ ⟨⟨ [gi, gyi ] i = 1, . . . , n, y ∈ G⟩⟩

The curve γ in the link complement, corresponding to the definingrelation [gi, g

yi ] of the Milnor group, becomes trivial after a

self-intersection of the component li:

gi

γy

If L, L′ are link-homotopic then their Milnor groups ML, ML′ areisomorphic.

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An important dichotomy:

Null-homotopic versus Homotopically essential links:

A link is null-homotopic if it is link-homotopic to the unlink.

Null-homotopic The Borromean Rings:

homotopically essential

Whitehead doubles of (null-homotopic)+ links are topologicallyslice. The slicing problem for Whitehead doubles of homotopicallyessential links (with linking numbers = 0) is a central problem,equivalent to the surgery conjecture.

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A link is null-homotopic if it is link-homotopic to the unlink.

Equivalently, a link is null-homotopic iff its components bounddisjoint maps of disks into D4.

Theorem. (Milnor) A link L = (l1, . . . , ln) is null-homotopic if andonly if its Milnor group is isomorphic to the free Milnor group,

ML ∼= MFm1,...,mn .

The free Milnor group is nilpotent of class n.

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Recall the defining relations of the Milnor group MG:[gi, g

yi ] where {gi} are a fixed set of normal generators.

2-Engel groups: groups satisfying the universal relation[x, [x, y]] = 1, or equivalently [x, xy] = 1.

In a 2-Engel group this relation applies to all group elements x, y,not just generators.

Drastic difference:

The free Milnor group Freeg1,...,gn/(Milnor relation) is nilpotent ofclass n.

Theorem (W. Burnside 1902, C. Hopkins 1929)Freeg1,...,gn/(2-Engel relation) is nilpotent of class 3.

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Freeg1,...,gn/(2-Engel relation) is nilpotent of class 3.

More generally: k-Engel groups satisfy the relation:[x, . . . , [x, [x, y]] . . .].

These higher relations may also turn out to be useful for thesurgery conjecture, but they are not well understood: It is animportant question in group theory whether k-Engel groups arenilpotent (k > 4).

A more relevant question: Does

Freeg1,...,gn/(k−Engel relation and the Milnor relation)

have a constant nilpotency class (independent of n)?

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y

x

y

x

(a) kinky handle (b) “secondary kinky handle”

(a) The double point implies [x, xy] = 1 or equivalently[x, [x, y]] = 1.

(b) The second double point corresponds to [x, [x, [x, [x, y]]]] = 1.

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Milnor relation ←→ link-homotopy

the 2-Engel relation ←→ weak homotopy of links

x

γy

An elementary weak homotopy.

The curve γ in the link complement, corresponding to the 2-Engelrelation [x, [x, y]] = [x, xy], becomes trivial after the move.

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Freeg1,...,gn/2-Engel relation is nilpotent of class 3.

It follows that any 4-fold commutator [x, [y, [z, w]]] is a product ofconjugates of the 2-Engel relation.

γ

x y z

w

γ = [x, [y, [z, w]]]

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Freeg1,...,gn/2-Engel relation is nilpotent of class 3.

Lemma. Let L be an n-component link in S3, and suppose thatML/(ML)5 ∼= MFn/(MFn)

5, or equivalently that all µ-invariantsof L with non-repeating coefficients of length ≤ 4 vanish. Then Lis weakly null-homotopic.

Note: The Milnor group is not functorial:

MG = G/ ⟨⟨ [gi, gyi ] i = 1, . . . , n, y ∈ G⟩⟩

Taking the quotient with respect to the 2-Engel relation [x, [x, y]]is functorial, but this kills most links!

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Examples of 2-Engel relations, and corresponding links:

γ1

x y

z w

γ1 = [x, [yz, [yz, w]]]

γ2

x zyw

γ2 = [x, [yw, [z, yw]]].

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A crucial feature of these basic weakly homotopic links: they arehomotopically essential, but after omitting one parallel componentthey become null-homotopic!

γ1

x y

z w

γ1 = [x, [yz, [yz, w]]]

γ1

x y

w

γ1 = [x, [y, [y, w]]]. Null-homotopic!

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γ2

x zyw

γ2 = [x, [yw, [z, yw]]].

γ2

x zy

γ2 = [x, [y, [z, y]]]. Null-homotopic!.

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II. 4-manifold topology

Motivation:

Geometric classification tools in higher dimensions:

Surgery: Given an n−dimensional Poincare complex X, is there ann−manifold Mn homotopy equivalent to it?

s-cobordism theorem: Given an (n+ 1)-dimensional s-cobordismW with ∂W = M1 ⊔ (−M2), is W isomorphic to the productM1 × [0, 1]?

In dimension n = 4: smoothly both surgery and s-cobordism faileven in the simply-connected case (Donaldson)

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Dimension n = 4, topological category:

M. Freedman (1982): Both surgery and s-cobordism conjectureshold for π1 = 1 and more generally for elementary amenablegroups.

Applications:

• Classification of topological simply-connected 4-manifolds.

• Slice results for knots and links, in particular: Alexanderpolynomial 1 knots are topologically slice.

• Classification of homeomorphisms (up to isotopy) ofsimply-connected 4-manifolds.

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The underlying technique:

Theorem (M.Freedman, 1982) The Casson handle ishomeomorphic to the standard 2-handle, D2 ×D2.

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The underlying technique:

Theorem (M.Freedman, 1982) The Casson handle ishomeomorphic to the standard 2-handle, D2 ×D2.

In the proof of both surgery and s-cobordisms theorems, the

question is whether a hyperbolic pair

(0 11 0

)in π2(M

4) may be

represented by embedded spheres:

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The class of good groups, for which surgery and the s-cobordismconjectures are known to hold, includes the groups ofsubexponential growth, and is closed under extensions and directlimits. (Freedman-Teichner 1995, K.-Quinn 2000)

Conjecture (Freedman 1983) Surgery fails for free groups.

More specifically, there does not exist a topological 4−manifoldM , homotopy equivalent to ∨3S1, with ∂M = S0(Wh(Bor)): thezero-framed surgery on the Whitehead double of the Borromeanrings.

Equivalently: The Whitehead double of the Borromean rings is nota “free” slice link.

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Figure: The untwisted Whitehead double of the Borromean rings.

Conjecture? The Whitehead double of the Borromean rings is nota “free” slice link.

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III. The A-B slice problem

Conjecture There does not exist a topological 4−manifold M ,homotopy equivalent to ∨3S1, with ∂M = S0(Wh(Bor)).

The A-B slice problem (Freedman ’86)

Suppose M4 exists. Its universal cover M is contractible. Theend-point compactification of M is homeomorphic to the 4−ball.π1(M), the free group on three generators, acts on D4.

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A1

A2A3

B1

B2B3

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A1

A2A3

B1

B2B3

A2

B2

B3

A3

A1B1

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A decomposition of D4, D4 = A ∪B, is an extension to the 4-ballof the standard genus one Heegaard decomposition of the3-sphere. Specified distinguished curves α ⊂ ∂A, β ⊂ ∂B form theHopf link in S3 = ∂D4.

α

α

A

β βB

A 2−dimensional example of a decomposition, D2 = A ∪B.

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Examples of decompositions. The trivial decomposition:

AB B

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Another example, D4 = A1 ∪B1:

A1

0

0B1

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A2

α

α

0

0 0

B2

β

β

0 0

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A3 B3

A4 B4

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An n−component link L ⊂ S3 is A−B slice if there existdecompositions (Ai, Bi), i = 1, . . . , n of D4 and disjointembeddings of all 2n manifolds {Ai, Bi} into D4 so that thedistinguished curves (α1, . . . , αn) form the link L, and the curves(β1, . . . , βn) form a parallel copy of L.

Moreover, the new embeddings Ai ⊂ D4, Bi ⊂ D4 are required tobe isotopic to the original embeddings.

Easy: The Hopf link is not A-B slice.

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Connection with the surgery conjecture:

Topological 4-dimensional surgery works for all groups if and onlyif the Borromean rings (and a certain family of theirgeneralizations) are A-B slice.

Freedman’s conjecture: The Borromean rings are not A-B slice.

A1

A2

A3

B1

B2B3

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The “relative-slice” problem. An illustration in 2 dimensions:

γ

γH∗

1

H2

M

δ δ

H′∗1

H′2

N

γ

γ

M

δ δ

H′∗1

ND′

Figure: Disjoint embeddings of (M,γ), (N, δ) in (D4, S3), where γ, δform a Hopf link in S3.

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Theorem (Freedman - K.) The generalized Borromean rings, acollection of links forming model surgery problems, are homotopyA-B slice.

An n-component link L is homotopy A-B slice if there existdecompositions D4 = Ai ∪Bi, i = 1, . . . , n and handledecompositions of the submanifolds Ai, Bi so that thecorresponding relative-slice problem has a link-homotopy solution.

In other words, all handles of Ai, Bi are mapped into D4 disjointly(but may have self-intersections).

Note: Much of the effort over the last 30 years was spent onsearch for a homotopy obstruction.

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l1

l2

l3

l4

l5

A link in the collection of universal surgery problems.

l1 is a 4-fold commutator, so it is a product of 2-Engel relations.

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0

Figure: Part A1 of the decomposition D4 = A1 ∪B1.

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The AB-slice problem has a solution for the Whitehead link. (TheWhitehead double of the Whitehead link is slice, Freedman ‘88.)More generally, there is a solution for (homotopically trivial)+ links(Freedman-Tiechner ‘95).

The new theorem reduces the universal surgery problems tohomotopically trivial links.

Question. Can these steps be combined to give a solution tosurgery?