Homotopy periods of link maps and Milnor’s
invariants
Clayton Shonkwiler
Department of Mathematics
University of Georgia
September 25, 2011
Joint work with Frederick R. Cohen and Rafal Komendarczyk
Links
Consider a parametrized n-component link L1 , L2 , L3 , . . . , Ln , where
Li : S 1 → R3
have disjoint images.
The Evaluation Map
Given an n-component link L = {L1 , . . . , Ln }, there is a natural
evaluation map
1
FL : S
. . × S }1 −→ Conf(n)
| × .{z
n times
which is just FL = L1 × . . . × Ln .
The Evaluation Map
Given an n-component link L = {L1 , . . . , Ln }, there is a natural
evaluation map
1
FL : S
. . × S }1 −→ Conf(n)
| × .{z
n times
which is just FL = L1 × . . . × Ln .
Here Conf(n) is the configuration space of n distinct points in R3 :
Conf(n) = Conf 3 R3 = {(x1 , . . . , xn ) ∈ (R3 )n : xi 6= xj for i 6= j}.
Link homotopy
Definition
A link homotopy of a link L is a deformation during which each
component may cross itself, but distinct components must remain
disjoint.
!
Link homotopy
Definition
A link homotopy of a link L is a deformation during which each
component may cross itself, but distinct components must remain
disjoint.
!
Let LM(n) denote the set of n-component links up to link
homotopy.
The κ invariant
FL : (S 1 )n −→ Conf(n).
Link homotopies of L induce homotopies of FL .
We get an induced map
κ : LM(n) −→ [(S 1 )n , Conf(n)]
L
7−→
[FL ]
n=2
S1 × S1
(t1 , t2 )
⊂
- Conf2 R3
- (L1 (t1 ), L2 (t2 ))
homotopy
equiv.
- S2
- L1 (t1 ) − L2 (t2 )
|L1 (t1 ) − L2 (t2 )|
Theorem (Gauss)
The degree of the composition is equal to the linking number.
n=2
S1 × S1
(t1 , t2 )
⊂
- Conf2 R3
- (L1 (t1 ), L2 (t2 ))
homotopy
equiv.
- S2
- L1 (t1 ) − L2 (t2 )
|L1 (t1 ) − L2 (t2 )|
Theorem (Gauss)
The degree of the composition is equal to the linking number.
Since the linking number classifies 2-component links up to link
homotopy and the degree classifies maps S 1 × S 1 → S 2 up to
homotopy, κ is bijective for n = 2.
n=3
S1 × S1 × S1
⊂
- Conf3 S 3 hom.
- S3 × S2
equiv.
π
- S 2.
n=3
S1 × S1 × S1
⊂
- Conf3 S 3 hom.
- S3 × S2
π
equiv.
- S 2.
z
y
x
S3
prz y
prz x
z⊥
−z
w = (prz x − prz y )/|prz x − prz y |
n=3
S1 × S1 × S1
⊂
- Conf3 S 3 hom.
- S3 × S2
equiv.
π
- S 2.
Theorem (with DeTurck, Gluck, Komendarczyk, Melvin, and
Vela-Vick)
The homotopy periods of the above map give the complete set of
link homotopy invariants of 3-component links.
n=3
S1 × S1 × S1
⊂
- Conf3 S 3 hom.
- S3 × S2
equiv.
π
- S 2.
Theorem (with DeTurck, Gluck, Komendarczyk, Melvin, and
Vela-Vick)
The homotopy periods of the above map give the complete set of
link homotopy invariants of 3-component links.
Corollary
κ : LM(3) → [(S 1 )3 , Conf(3)] is injective (though not bijective).
Koschorke’s Conjecture
Conjecture (Koschorke)
The map κ : LM(n) → [(S 1 )n , Conf(n)] is injective for all n.
Koschorke’s Conjecture
Conjecture (Koschorke)
The map κ : LM(n) → [(S 1 )n , Conf(n)] is injective for all n.
Let BLM(n) denote the set of homotopy Brunnian n-component
links, meaning every (n − 1)-component sublink is link
homotopically trivial.
Koschorke’s Conjecture
Conjecture (Koschorke)
The map κ : LM(n) → [(S 1 )n , Conf(n)] is injective for all n.
Let BLM(n) denote the set of homotopy Brunnian n-component
links, meaning every (n − 1)-component sublink is link
homotopically trivial.
Theorem (Koschorke)
The restriction κ : BLM(n) → [(S 1 )n , Conf(n)] is injective.
Milnor’s invariants
Let L be an n-component link and let I = (i1 , . . . , im ) be a
multi-index with 1 ≤ ik ≤ n for all k.
Milnor’s invariants
Let L be an n-component link and let I = (i1 , . . . , im ) be a
multi-index with 1 ≤ ik ≤ n for all k.
Milnor defined invariants µ(I ; j), which are integers modulo the
greatest common divisor of the lower-order invariants µ(J; j) with
J a proper subset of I . These are link homotopy invariants when
the ik are all distinct from each other and from j.
Milnor’s invariants
Let L be an n-component link and let I = (i1 , . . . , im ) be a
multi-index with 1 ≤ ik ≤ n for all k.
Milnor defined invariants µ(I ; j), which are integers modulo the
greatest common divisor of the lower-order invariants µ(J; j) with
J a proper subset of I . These are link homotopy invariants when
the ik are all distinct from each other and from j.
Theorem (Milnor)
Elements of BLM(n) are completely classified by the µ(I ; j) with
|I | = n − 1 and I ∩ {j} = ∅. Moreover, all lower-order µ invariants
vanish for Brunnian links, so these invariants are integers.
An Example
An Example
•
µ(i; j) = Lk(Li , Lj ) = 0 for all i 6= j.
•
µ(1, 2; 3) = µ(2, 3; 1) = µ(3, 1; 2) = 1
µ(2, 1; 3) = µ(3, 2; 1) = µ(1, 3; 2) = −1.
Homotopy Periods and Milnor’s Invariants
Theorem (with Cohen and Komendarczyk)
Let {Bj,i : 1 ≤ i < j ≤ n} be the generators of the Lie algebra
π∗ (ΩConf(n)) ⊗ Q.
Homotopy Periods and Milnor’s Invariants
Theorem (with Cohen and Komendarczyk)
Let {Bj,i : 1 ≤ i < j ≤ n} be the generators of the Lie algebra
π∗ (ΩConf(n)) ⊗ Q. Then
1
κ(BLM(n)) ⊂ πn (Conf(n)) ⊂ [(S 1 )n , Conf(n)] and is a Lie
Z-module generated by the iterated Samelson products
B(n, σ) = [Bn,1 , Bn,σ(2) , . . . , Bn,σ(n−1) ]
for σ ∈ Σ(2, . . . , n − 1).
Homotopy Periods and Milnor’s Invariants
Theorem (with Cohen and Komendarczyk)
Let {Bj,i : 1 ≤ i < j ≤ n} be the generators of the Lie algebra
π∗ (ΩConf(n)) ⊗ Q. Then
1
κ(BLM(n)) ⊂ πn (Conf(n)) ⊂ [(S 1 )n , Conf(n)] and is a Lie
Z-module generated by the iterated Samelson products
B(n, σ) = [Bn,1 , Bn,σ(2) , . . . , Bn,σ(n−1) ]
for σ ∈ Σ(2, . . . , n − 1).
2
For any L ∈ BLM(n),
X
κ([L]) =
σ∈Σ(2,...,n−1)
µ(1, σ(2), . . . , σ(n − 1); n)B(n, σ)
Homotopy Periods and Milnor’s Invariants
Theorem (with Cohen and Komendarczyk)
Let {Bj,i : 1 ≤ i < j ≤ n} be the generators of the Lie algebra
π∗ (ΩConf(n)) ⊗ Q. Then
1
κ(BLM(n)) ⊂ πn (Conf(n)) ⊂ [(S 1 )n , Conf(n)] and is a Lie
Z-module generated by the iterated Samelson products
B(n, σ) = [Bn,1 , Bn,σ(2) , . . . , Bn,σ(n−1) ]
for σ ∈ Σ(2, . . . , n − 1).
2
For any L ∈ BLM(n),
X
κ([L]) =
µ(1, σ(2), . . . , σ(n − 1); n)B(n, σ)
σ∈Σ(2,...,n−1)
In other words, the homotopy periods of FL are given by the
(n − 2)! invariants {µ(1, σ(2), . . . , σ(n − 1); n)}σ .
String Links
Habegger and Lin introduced the group of homotopy string links
H (n).
String Links
Habegger and Lin introduced the group of homotopy string links
H (n).
String Links
Habegger and Lin introduced the group of homotopy string links
H (n).
Theorem (Habegger–Lin)
The map H (n) → LM(n) induced by the Markov closure is
surjective. Moreover, if two string links close up to link-homotopic
links, then they are related by a sequence of conjugations and
“partial conjugations”.
Brunnian String Links
For each i = 1, . . . , n, there is a natural projection
δi : H (n) → H (n − 1)
given by deleting the ith strand.
Brunnian String Links
For each i = 1, . . . , n, there is a natural projection
δi : H (n) → H (n − 1)
given by deleting the ith strand.
BH (n) = ker
Y
is the subgroup of Brunnian string links.
Key Proposition 1
BH (n) is the center of H (n).
δi
Brunnian String Links
For each i = 1, . . . , n, there is a natural projection
δi : H (n) → H (n − 1)
given by deleting the ith strand.
BH (n) = ker
Y
δi
is the subgroup of Brunnian string links.
Key Proposition 1
BH (n) is the center of H (n).
Corollary
The restriction of the Markov closure map BH (n) → BLM(n) is
bijective.
Torus Homotopy Groups
Let T (n) be the nth torus homotopy group [Fox] of Conf(n):
T (n) = [Σ(S 1 )n−1 , Conf(n)] = [(S 1 )n−1 , ΩConf(n)].
Torus Homotopy Groups
Let T (n) be the nth torus homotopy group [Fox] of Conf(n):
T (n) = [Σ(S 1 )n−1 , Conf(n)] = [(S 1 )n−1 , ΩConf(n)].
If L ∈ BLM(n), then L − Ln is link homotopically trivial, so the
restriction of FL to the face (S 1 )n−1 × {∗} is homotopic to the
constant map.
Torus Homotopy Groups
Let T (n) be the nth torus homotopy group [Fox] of Conf(n):
T (n) = [Σ(S 1 )n−1 , Conf(n)] = [(S 1 )n−1 , ΩConf(n)].
If L ∈ BLM(n), then L − Ln is link homotopically trivial, so the
restriction of FL to the face (S 1 )n−1 × {∗} is homotopic to the
constant map.
Therefore, we can interpret κ(L) = [FL ] as an element of T (n).
Injectivity
Key Proposition 2
The diagram
φ
BH (n)
- T (n)
p#
?
BLM(n)
κ
?
- [(S 1 )n , Conf(n)]
commutes and p # ◦ φ is injective. Therefore κ is injective as well.
Thanks!