25th Nordic and 1st British-Nordic congress of Mathematicians

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Definitions and constructions Homotopy theory Maps Golod

25th Nordic and 1st British-Nordic congress of Mathematicians

The homotopy theory of moment-angle complexes Jelena Grbi´ c

University of Manchester

joint work with Stephen Theriualt (University of Aberdeen) June 2009

The homotopy theory of moment-angle complexes Jelena Grbi´c 25th Nordic and 1st British-Nordic congress of Mathematicians

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TORIC TOPOLOGY from a homotopy theoretical point of view

The homotopy type of the complement of a coordinate subspace

arrangement

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TORIC TOPOLOGY from a homotopy theoretical point of view

The homotopy type of the complement of a coordinate subspace arrangement

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Combinatorics: Simplicial complex

V = {v ₁ , . . . , v _n } = [n] set of vertices Definition

An abstract simplicial complex on V is

K := {σ ₁ , . . . , σ _s | σ _i ⊂ V } ( ∅ ∈ K ) closed under formation of subsets.

σ ∈ K – simplex dim(σ) = |σ| − 1

dim(K ) = max σ∈K {dim(σ)}

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Algebra: Stanley-Reisner face ring

R – commutative ring with unit deg(v _i ) = 2 – topological grading

R[V ] = R[v ₁ , . . . , v _n ] graded polynomial algebra on V over R

Given σ = {i ₁ , . . . , i r } ⊂ [n], set

v ^σ = v _i

₁

. . . v _i

_r

.

Definition

The Stanley-Reisner algebra (or face ring) of K is R[K ] := R[v 1 , . . . , v n ]/(v ^σ | σ / ∈ K ).

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Algebra: Stanley-Reisner face ring

R – commutative ring with unit deg(v _i ) = 2 – topological grading

R[V ] = R[v ₁ , . . . , v _n ] graded polynomial algebra on V over R Given σ = {i ₁ , . . . , i r } ⊂ [n], set

v ^σ = v _i

₁

. . . v _i

_r

.

Definition

The Stanley-Reisner algebra (or face ring) of K is

R[K ] := R[v 1 , . . . , v n ]/(v ^σ | σ / ∈ K ).

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Algebra: Stanley-Reisner face ring

R – commutative ring with unit deg(v _i ) = 2 – topological grading

R[V ] = R[v ₁ , . . . , v _n ] graded polynomial algebra on V over R Given σ = {i ₁ , . . . , i r } ⊂ [n], set

v ^σ = v _i

₁

. . . v _i

_r

.

Definition

The Stanley-Reisner algebra (or face ring) of K is R[K ] := R[v 1 , . . . , v n ]/(v ^σ | σ / ∈ K ).

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Topology: Polyhedral products (X , A) ^K

Let (X , A) := (X ₁ , A ₁ ), . . . , (X _n , A _n ) .

Given σ ⊂ [n], define (X , A) ^σ =

(x ₁ , . . . , x _n ) ∈ Q n

i=1 X _i | x _i ∈ A _i for i ∈ / σ .

Definition

For K on [n], define the polyhedral product (X , A) ^K by (X , A) ^K := [

σ∈K

(X , A) ^σ .

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Topology: Polyhedral products (X , A) ^K

Let (X , A) := (X ₁ , A ₁ ), . . . , (X _n , A _n ) . Given σ ⊂ [n], define

(X , A) ^σ =

(x ₁ , . . . , x _n ) ∈ Q n

i=1 X _i | x _i ∈ A _i for i ∈ / σ .

Definition

For K on [n], define the polyhedral product (X , A) ^K by (X , A) ^K := [

σ∈K

(X , A) ^σ .

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Topology: Polyhedral products (X , A) ^K

Let (X , A) := (X ₁ , A ₁ ), . . . , (X _n , A _n ) . Given σ ⊂ [n], define

(X , A) ^σ =

(x ₁ , . . . , x _n ) ∈ Q n

i=1 X _i | x _i ∈ A _i for i ∈ / σ .

Definition

For K on [n], define the polyhedral product (X , A) ^K by (X , A) ^K := [

σ∈K

(X , A) ^σ .

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Topology: Moment-angle complex Z _K

Definition

The moment–angle complex is given by Z _K := (D ² , S ¹ ) ^K .

(D ² , S ¹ ) ^σ invariant under the action of T ⁿ T ⁿ acts on Z _K Lemma

Z _K /T ⁿ ∼ = ConeK ⁰ K ⁰ - barycentric subdivision of K . Proposition

If K is a triangulation of an (l − 1)-sphere, then Z _K is a closed (l + n)-manifold.

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Topology: Moment-angle complex Z _K

Definition

The moment–angle complex is given by Z _K := (D ² , S ¹ ) ^K .

(D ² , S ¹ ) ^σ invariant under the action of T ⁿ T ⁿ acts on Z _K

Lemma

Z _K /T ⁿ ∼ = ConeK ⁰ K ⁰ - barycentric subdivision of K . Proposition

If K is a triangulation of an (l − 1)-sphere, then Z _K is a closed

(l + n)-manifold.

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Topology: Moment-angle complex Z _K

Definition

The moment–angle complex is given by Z _K := (D ² , S ¹ ) ^K .

(D ² , S ¹ ) ^σ invariant under the action of T ⁿ T ⁿ acts on Z _K Lemma

Z _K /T ⁿ ∼ = ConeK ⁰ K ⁰ - barycentric subdivision of K .

Proposition

If K is a triangulation of an (l − 1)-sphere, then Z _K is a closed (l + n)-manifold.

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Topology: Moment-angle complex Z _K

Definition

The moment–angle complex is given by Z _K := (D ² , S ¹ ) ^K .

(D ² , S ¹ ) ^σ invariant under the action of T ⁿ T ⁿ acts on Z _K Lemma

Z _K /T ⁿ ∼ = ConeK ⁰ K ⁰ - barycentric subdivision of K . Proposition

If K is a triangulation of an (l − 1)-sphere, then Z _K is a closed

(l + n)-manifold.

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Example

If n = 2, and K = {v ₁ , v 2 }, then

Z _K ∼ = S ³ ∼ = D ² × S ¹ ∪ S ¹ × D ² ⊂ D ² × D ² .

Example

Let K = ∂∆ ⁿ⁻¹ , then Z _K = ∂((D ² ) ⁿ ) ∼ = S ²ⁿ⁻¹

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Example

If n = 2, and K = {v ₁ , v 2 }, then

Z _K ∼ = S ³ ∼ = D ² × S ¹ ∪ S ¹ × D ² ⊂ D ² × D ² . Example

Let K = ∂∆ ⁿ⁻¹ , then Z _K = ∂((D ² ) ⁿ ) ∼ = S ²ⁿ⁻¹

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Topology: Davis–Januszkiewicz space DJ _K

Davis and Januszkiewicz consider the homotopy orbit space DJ _K := ET ⁿ × _T

ⁿ

Z _K .

Definition (Buchstaber-Panov)

The Davis–Januszkiewicz space is given by DJ _K := (BT , ∗) ^K . Proposition

H ^∗ (DJ _K ; Z ) = Z [K ], H _T ^∗

n

(Z _K ) = Z [K ] Proposition

There is a homotopy fibration

Z _K −→ DJ _K −→ ⁱ BT ⁿ . (D ² , S ¹ ) ^K −→ (BT , ∗) ^K −→ (BT , BT ) ^K

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Topology: Davis–Januszkiewicz space DJ _K

Davis and Januszkiewicz consider the homotopy orbit space DJ _K := ET ⁿ × _T

ⁿ

Z _K .

Definition (Buchstaber-Panov)

The Davis–Januszkiewicz space is given by DJ _K := (BT , ∗) ^K .

Proposition

H ^∗ (DJ _K ; Z ) = Z [K ], H _T ^∗

n

(Z _K ) = Z [K ] Proposition

There is a homotopy fibration

Z _K −→ DJ _K −→ ⁱ BT ⁿ .

(D ² , S ¹ ) ^K −→ (BT , ∗) ^K −→ (BT , BT ) ^K

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Topology: Davis–Januszkiewicz space DJ _K

Davis and Januszkiewicz consider the homotopy orbit space DJ _K := ET ⁿ × _T

ⁿ

Z _K .

Definition (Buchstaber-Panov)

The Davis–Januszkiewicz space is given by DJ _K := (BT , ∗) ^K . Proposition

H ^∗ (DJ _K ; Z ) = Z [K ], H _T ^∗

n

(Z _K ) = Z [K ]

Proposition

There is a homotopy fibration

Z _K −→ DJ _K −→ ⁱ BT ⁿ . (D ² , S ¹ ) ^K −→ (BT , ∗) ^K −→ (BT , BT ) ^K

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Topology: Davis–Januszkiewicz space DJ _K

Davis and Januszkiewicz consider the homotopy orbit space DJ _K := ET ⁿ × _T

ⁿ

Z _K .

Definition (Buchstaber-Panov)

The Davis–Januszkiewicz space is given by DJ _K := (BT , ∗) ^K . Proposition

H ^∗ (DJ _K ; Z ) = Z [K ], H _T ^∗

n

(Z _K ) = Z [K ] Proposition

There is a homotopy fibration Z _K −→ DJ K

−→ i BT ⁿ .

(D ² , S ¹ ) ^K −→ (BT , ∗) ^K −→ (BT , BT ) ^K

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Combinatorial geometry: Arrangements

Given σ = {i ₁ , . . . , i _k } ⊂ [n], define the coordinate subspace L _σ =

(z ₁ , . . . , z _n ) ∈ C ⁿ | z _i

₁

= . . . = z _i

_k

= 0 .

Definition

For K on [n], define the complex coordinate subspace arrangement CA(K ) :=

L σ | σ / ∈ K . Its complement in C ⁿ is given by U _K := C ⁿ \ S

σ / ∈K L σ .

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Combinatorial geometry: Arrangements

Given σ = {i ₁ , . . . , i _k } ⊂ [n], define the coordinate subspace L _σ =

(z ₁ , . . . , z _n ) ∈ C ⁿ | z _i

₁

= . . . = z _i

_k

= 0 .

Definition

For K on [n], define the complex coordinate subspace arrangement CA(K ) :=

L σ | σ / ∈ K . Its complement in C ⁿ is given by U _K := C ⁿ \ S

σ / ∈K L σ .

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Theorem (Buchstaber–Panov)

There is an equivariant deformation retraction U K

−→ Z ' _K .

Notice U K = (C, C ^∗ ) ^K .

( C , C ^∗ ) ^K −→ (D ² , S ¹ ) ^K Theorem

The following isomorphism of graded algebra holds H ^∗ (U K ; Z) ∼ = Tor _Z[v

₁

_,...,v

_n

_] (Z[K ], Z) ∼ = M

p ≥ −1

ω ⊂ [m]

H ^p (K ω ).

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Theorem (Buchstaber–Panov)

There is an equivariant deformation retraction U K

−→ Z ' _K .

Notice U K = (C, C ^∗ ) ^K .

( C , C ^∗ ) ^K −→ (D ² , S ¹ ) ^K Theorem

The following isomorphism of graded algebra holds H ^∗ (U K ; Z) ∼ = Tor _Z[v

₁

_,...,v

_n

_] (Z[K ], Z) ∼ = M

p ≥ −1

ω ⊂ [m]

H ^p (K ω ).

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Theorem (Buchstaber–Panov)

There is an equivariant deformation retraction U K

−→ Z ' _K .

Notice U K = (C, C ^∗ ) ^K .

( C , C ^∗ ) ^K −→ (D ² , S ¹ ) ^K

Theorem

The following isomorphism of graded algebra holds H ^∗ (U K ; Z) ∼ = Tor _Z[v

₁

_,...,v

_n

_] (Z[K ], Z) ∼ = M

p ≥ −1

ω ⊂ [m]

H ^p (K ω ).

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Theorem (Buchstaber–Panov)

There is an equivariant deformation retraction U K

−→ Z ' _K .

Notice U K = (C, C ^∗ ) ^K .

( C , C ^∗ ) ^K −→ (D ² , S ¹ ) ^K Theorem

The following isomorphism of graded algebra holds H ^∗ (U K ; Z) ∼ = Tor _Z[v

₁

_,...,v

_n

_] (Z[K ], Z) ∼ = M

p ≥ −1

ω ⊂ [m]

H ^p (K ω ).

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Unstable homotopy type of Z _K

Definition

A simplicial complex K is shifted if there is an ordering σ ∈ K , v ⁰ < v ⇒ (σ − v) ∪ v ⁰ ∈ K .

Proposition

If K is shifted, then all Massey products in H ^∗ (Z _K ) vanish. Strategy: Study the homotopy fibration

Z _K = (D ² , S ¹ ) ^K −→ DJ _K = (BT , ∗) ^K −→ BT ⁿ . We actually consider the generalise fibration

(Cone ΩX , ΩX ) ^K −→ (X , ∗) ^K −→ Y X .

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Unstable homotopy type of Z _K

Definition

A simplicial complex K is shifted if there is an ordering σ ∈ K , v ⁰ < v ⇒ (σ − v) ∪ v ⁰ ∈ K . Proposition

If K is shifted, then all Massey products in H ^∗ (Z _K ) vanish.

Strategy: Study the homotopy fibration

Z _K = (D ² , S ¹ ) ^K −→ DJ _K = (BT , ∗) ^K −→ BT ⁿ . We actually consider the generalise fibration

(Cone ΩX , ΩX ) ^K −→ (X , ∗) ^K −→ Y

X .

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Unstable homotopy type of Z _K

Definition

A simplicial complex K is shifted if there is an ordering σ ∈ K , v ⁰ < v ⇒ (σ − v) ∪ v ⁰ ∈ K . Proposition

If K is shifted, then all Massey products in H ^∗ (Z _K ) vanish.

Strategy: Study the homotopy fibration

Z _K = (D ² , S ¹ ) ^K −→ DJ _K = (BT , ∗) ^K −→ BT ⁿ .

We actually consider the generalise fibration (Cone ΩX , ΩX ) ^K −→ (X , ∗) ^K −→ Y

X .

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Unstable homotopy type of Z _K

Definition

A simplicial complex K is shifted if there is an ordering σ ∈ K , v ⁰ < v ⇒ (σ − v) ∪ v ⁰ ∈ K . Proposition

If K is shifted, then all Massey products in H ^∗ (Z _K ) vanish.

Strategy: Study the homotopy fibration

Z _K = (D ² , S ¹ ) ^K −→ DJ _K = (BT , ∗) ^K −→ BT ⁿ . We actually consider the generalise fibration

(Cone ΩX , ΩX ) ^K −→ (X , ∗) ^K −→ Y

X .

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Theorem (G., Theriault)

Let K be a shifted complex. Then (Cone ΩX , ΩX ) ^K ' _

1 ≤ t < n (i 1 , . . . , i k )

Σ ^t ΩX _i

₁

∧ . . . ∧ ΩX _i

_k

.

In particular, Z _K is a wedge of spheres.

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Let X ₁ , . . . , X _n be path-connected spaces.

There is a filtration of X ₁ × . . . × X _n given by T _n ⁿ −→ T _n−1 ⁿ −→ · · · −→ T ₀ ⁿ where T _k ⁿ =

(x 1 , . . . , x n ) ∈ X 1 × . . . × X n | at least k of x i ‘s are ∗ .

Notice T _k ⁿ ( C P ^∞ ) = DJ _∆

^n−k−1

_[n] .

Denote by F _k ⁿ (X ) the homotopy fibre of T _k ⁿ −→ Q n i=1 X _i . For X i = CP ^∞ , F _k ⁿ (CP ^∞ ) = Z _∆

n−k−1

[n]

Theorem (Porter; G., Theriault)

For n ≥ 1, and k such that 1 ≤ k ≤ n − 1, F _k ⁿ '

n

_

j =n−k +1

_

1≤i

₁

<...<i

j

≤n

j − 1 n − k

Σ ^n−k ΩX _i

₁

∧ . . . ∧ ΩX _i

_j

F _k ⁿ (CP ^∞ ) '

n

_

j =n−k +1

n j

j − 1 n − k

S ^n+j−k .

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Let X ₁ , . . . , X _n be path-connected spaces.

There is a filtration of X ₁ × . . . × X _n given by T _n ⁿ −→ T _n−1 ⁿ −→ · · · −→ T ₀ ⁿ where T _k ⁿ =

(x 1 , . . . , x n ) ∈ X 1 × . . . × X n | at least k of x i ‘s are ∗ . Notice T _k ⁿ ( C P ^∞ ) = DJ _∆

^n−k−1

_[n] .

Denote by F _k ⁿ (X ) the homotopy fibre of T _k ⁿ −→ Q n i=1 X _i . For X i = CP ^∞ , F _k ⁿ (CP ^∞ ) = Z _∆

n−k−1

[n]

Theorem (Porter; G., Theriault)

For n ≥ 1, and k such that 1 ≤ k ≤ n − 1, F _k ⁿ '

n

_

j =n−k +1

_

1≤i

₁

<...<i

j

≤n

j − 1 n − k

Σ ^n−k ΩX _i

₁

∧ . . . ∧ ΩX _i

_j

F _k ⁿ (CP ^∞ ) '

n

_

j =n−k +1

n j

j − 1 n − k

S ^n+j−k .

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Let X ₁ , . . . , X _n be path-connected spaces.

There is a filtration of X ₁ × . . . × X _n given by T _n ⁿ −→ T _n−1 ⁿ −→ · · · −→ T ₀ ⁿ where T _k ⁿ =

(x 1 , . . . , x n ) ∈ X 1 × . . . × X n | at least k of x i ‘s are ∗ . Notice T _k ⁿ ( C P ^∞ ) = DJ _∆

^n−k−1

_[n] .

Denote by F _k ⁿ (X ) the homotopy fibre of T _k ⁿ −→ Q n i=1 X _i . For X i = CP ^∞ , F _k ⁿ (CP ^∞ ) = Z _∆

n−k−1

[n]

Theorem (Porter; G., Theriault)

For n ≥ 1, and k such that 1 ≤ k ≤ n − 1, F _k ⁿ '

n

_

j =n−k +1

_

1≤i

₁

<...<i

j

≤n

j − 1 n − k

Σ ^n−k ΩX _i

₁

∧ . . . ∧ ΩX _i

_j

F _k ⁿ (CP ^∞ ) '

n

_

j =n−k +1

n j

j − 1 n − k

S ^n+j−k .

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Let X ₁ , . . . , X _n be path-connected spaces.

There is a filtration of X ₁ × . . . × X _n given by T _n ⁿ −→ T _n−1 ⁿ −→ · · · −→ T ₀ ⁿ where T _k ⁿ =

(x 1 , . . . , x n ) ∈ X 1 × . . . × X n | at least k of x i ‘s are ∗ . Notice T _k ⁿ ( C P ^∞ ) = DJ _∆

^n−k−1

_[n] .

Denote by F _k ⁿ (X ) the homotopy fibre of T _k ⁿ −→ Q n i=1 X _i . For X i = CP ^∞ , F _k ⁿ (CP ^∞ ) = Z _∆

n−k−1

[n]

Theorem (Porter; G., Theriault)

For n ≥ 1, and k such that 1 ≤ k ≤ n − 1, F _k ⁿ '

n

_

j =n−k +1

_

1≤i

₁

<...<i

j

≤n

j − 1 n − k

Σ ^n−k ΩX _i

₁

∧ . . . ∧ ΩX _i

_j

F _k ⁿ (CP ^∞ ) '

n

_

j =n−k +1

n j

j − 1 n − k

S ^n+j−k .

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Higher Whitehead products and fat wedges

For X = {X ₁ , . . . , X n } the fat wedge is the space

FW (X ) = {(x ₁ , . . . , x n ) ∈ X 1 × · · · × X n | at least one x i = ∗}.

Specialise X = ΣY . There is a homotopy fibration

Σ ⁿ⁻¹ ΩΣY 1 ∧ · · · ∧ ΩΣY n −→ FW (ΣY ) −→ ΣY 1 × · · · × ΣY n . The suspension map E : Y −→ ΩΣY induces a composite

φ n : Σ ⁿ⁻¹ Y 1 ∧ · · · ∧ Y n −→ Σ ⁿ⁻¹ ΩΣY 1 ∧ · · · ∧ ΩΣY n −→ FW (ΣY ). There is a homotopy cofibration

Σ ⁿ⁻¹ Y 1 ∧ · · · ∧ Y n φ

n

−→ FW (ΣY ) −→ ΣY 1 × · · · × ΣY n .

If n = 2, then φ 2 = [i 1 , i 2 ] : ΣY 1 ∧ Y 2 −→ ΣY 1 ∨ ΣY 2 is the

universal example for Whitehead products.

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Higher Whitehead products and fat wedges

For X = {X ₁ , . . . , X n } the fat wedge is the space

FW (X ) = {(x ₁ , . . . , x n ) ∈ X 1 × · · · × X n | at least one x i = ∗}.

Specialise X = ΣY . There is a homotopy fibration

Σ ⁿ⁻¹ ΩΣY 1 ∧ · · · ∧ ΩΣY n −→ FW (ΣY ) −→ ΣY 1 × · · · × ΣY n .

The suspension map E : Y −→ ΩΣY induces a composite

φ n : Σ ⁿ⁻¹ Y 1 ∧ · · · ∧ Y n −→ Σ ⁿ⁻¹ ΩΣY 1 ∧ · · · ∧ ΩΣY n −→ FW (ΣY ). There is a homotopy cofibration

Σ ⁿ⁻¹ Y 1 ∧ · · · ∧ Y n φ

n

−→ FW (ΣY ) −→ ΣY 1 × · · · × ΣY n . If n = 2, then φ 2 = [i 1 , i 2 ] : ΣY 1 ∧ Y 2 −→ ΣY 1 ∨ ΣY 2 is the universal example for Whitehead products.

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Higher Whitehead products and fat wedges

For X = {X ₁ , . . . , X n } the fat wedge is the space

FW (X ) = {(x ₁ , . . . , x n ) ∈ X 1 × · · · × X n | at least one x i = ∗}.

Specialise X = ΣY . There is a homotopy fibration

Σ ⁿ⁻¹ ΩΣY 1 ∧ · · · ∧ ΩΣY n −→ FW (ΣY ) −→ ΣY 1 × · · · × ΣY n . The suspension map E : Y −→ ΩΣY induces a composite

φ n : Σ ⁿ⁻¹ Y 1 ∧ · · · ∧ Y n −→ Σ ⁿ⁻¹ ΩΣY 1 ∧ · · · ∧ ΩΣY n −→ FW (ΣY ).

There is a homotopy cofibration Σ ⁿ⁻¹ Y 1 ∧ · · · ∧ Y n

φ

n

−→ FW (ΣY ) −→ ΣY 1 × · · · × ΣY n .

If n = 2, then φ 2 = [i 1 , i 2 ] : ΣY 1 ∧ Y 2 −→ ΣY 1 ∨ ΣY 2 is the

universal example for Whitehead products.

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Higher Whitehead products and fat wedges

For X = {X ₁ , . . . , X n } the fat wedge is the space

FW (X ) = {(x ₁ , . . . , x n ) ∈ X 1 × · · · × X n | at least one x i = ∗}.

Specialise X = ΣY . There is a homotopy fibration

Σ ⁿ⁻¹ ΩΣY 1 ∧ · · · ∧ ΩΣY n −→ FW (ΣY ) −→ ΣY 1 × · · · × ΣY n . The suspension map E : Y −→ ΩΣY induces a composite

φ n : Σ ⁿ⁻¹ Y 1 ∧ · · · ∧ Y n −→ Σ ⁿ⁻¹ ΩΣY 1 ∧ · · · ∧ ΩΣY n −→ FW (ΣY ).

There is a homotopy cofibration Σ ⁿ⁻¹ Y 1 ∧ · · · ∧ Y n

φ

n

−→ FW (ΣY ) −→ ΣY 1 × · · · × ΣY n .

If n = 2, then φ 2 = [i 1 , i 2 ] : ΣY 1 ∧ Y 2 −→ ΣY 1 ∨ ΣY 2 is the universal example for Whitehead products.

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Higher Whitehead products and fat wedges

For X = {X ₁ , . . . , X n } the fat wedge is the space

FW (X ) = {(x ₁ , . . . , x n ) ∈ X 1 × · · · × X n | at least one x i = ∗}.

Specialise X = ΣY . There is a homotopy fibration

Σ ⁿ⁻¹ ΩΣY 1 ∧ · · · ∧ ΩΣY n −→ FW (ΣY ) −→ ΣY 1 × · · · × ΣY n . The suspension map E : Y −→ ΩΣY induces a composite

φ n : Σ ⁿ⁻¹ Y 1 ∧ · · · ∧ Y n −→ Σ ⁿ⁻¹ ΩΣY 1 ∧ · · · ∧ ΩΣY n −→ FW (ΣY ).

There is a homotopy cofibration Σ ⁿ⁻¹ Y 1 ∧ · · · ∧ Y n

φ

n

−→ FW (ΣY ) −→ ΣY 1 × · · · × ΣY n .

If n = 2, then φ 2 = [i 1 , i 2 ] : ΣY 1 ∧ Y 2 −→ ΣY 1 ∨ ΣY 2 is the

universal example for Whitehead products.

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Take maps f i : ΣY i −→ Z and let their wedge sum W _n

i=1 ΣY _i −→ Z extends to f : FW (ΣY ) −→ Z . Definition

The n ^th -higher Whitehead product of the maps f 1 , . . . , f n is [f ₁ , . . . , f _n ] : Σ ⁿ⁻¹ Y ₁ ∧ · · · ∧ Y _n −→ ^φ

ⁿ

FW (ΣY ) −→ ^f Z .

If σ = (i 1 , . . . , i k ), let FW σ (ΣY ) be the fat wedge of Q k

j =1 ΣY i

j

. Lemma

If σ = (i 1 , . . . , i _k ) is a missing face of K , then there is a map FW _σ (ΣY ) −→ DJ _K (ΣY ) and the composite

Σ ^k−1 Y _i

₁

∧ · · · ∧ Y _i

_k

−→ ^φ

^k

FW _σ (ΣY ) −→ DJ _K (ΣY ) is nontrivial and it lifts to Z _K (ΣY ).

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Take maps f i : ΣY i −→ Z and let their wedge sum W _n

i=1 ΣY _i −→ Z extends to f : FW (ΣY ) −→ Z . Definition

The n ^th -higher Whitehead product of the maps f 1 , . . . , f n is [f ₁ , . . . , f _n ] : Σ ⁿ⁻¹ Y ₁ ∧ · · · ∧ Y _n −→ ^φ

ⁿ

FW (ΣY ) −→ ^f Z . If σ = (i 1 , . . . , i k ), let FW σ (ΣY ) be the fat wedge of Q k

j =1 ΣY i

j

.

Lemma

If σ = (i 1 , . . . , i _k ) is a missing face of K , then there is a map FW _σ (ΣY ) −→ DJ _K (ΣY ) and the composite

Σ ^k−1 Y _i

₁

∧ · · · ∧ Y _i

_k

−→ ^φ

^k

FW _σ (ΣY ) −→ DJ _K (ΣY )

is nontrivial and it lifts to Z _K (ΣY ).

(43)

Take maps f i : ΣY i −→ Z and let their wedge sum W _n

i=1 ΣY _i −→ Z extends to f : FW (ΣY ) −→ Z . Definition

The n ^th -higher Whitehead product of the maps f 1 , . . . , f n is [f ₁ , . . . , f _n ] : Σ ⁿ⁻¹ Y ₁ ∧ · · · ∧ Y _n −→ ^φ

ⁿ

FW (ΣY ) −→ ^f Z . If σ = (i 1 , . . . , i k ), let FW σ (ΣY ) be the fat wedge of Q k

j =1 ΣY i

j

. Lemma

If σ = (i 1 , . . . , i _k ) is a missing face of K , then there is a map FW _σ (ΣY ) −→ DJ _K (ΣY ) and the composite

Σ ^k−1 Y _i

₁

∧ · · · ∧ Y _i

_k

−→ ^φ

^k

FW _σ (ΣY ) −→ DJ _K (ΣY ) is nontrivial and it lifts to Z _K (ΣY ).

(44)

MF -complexes

Definition

A simplicial complex K is an MF -complex if

|K | = [

σ∈MF (K)

|∂σ|.

If K is an MF -complex, then

DJ _K (X ) = colim

σ∈MF (K) FW (σ).

Theorem

Let K be an MF -complex on [n]. Then each of Z _K (S) and Z _K is

homotopy equivalent to a wedge of spheres.

(45)

MF -complexes

Definition

A simplicial complex K is an MF -complex if

|K | = [

σ∈MF (K)

|∂σ|.

If K is an MF -complex, then

DJ _K (X ) = colim

σ∈MF (K ) FW (σ).

Theorem

Let K be an MF -complex on [n]. Then each of Z _K (S) and Z _K is homotopy equivalent to a wedge of spheres.

(46)

MF -complexes

Definition

A simplicial complex K is an MF -complex if

|K | = [

σ∈MF (K)

|∂σ|.

If K is an MF -complex, then

DJ _K (X ) = colim

σ∈MF (K ) FW (σ).

Theorem

Let K be an MF -complex on [n]. Then each of Z _K (S) and Z _K is

homotopy equivalent to a wedge of spheres.

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Strategy

calculate the rational loop homology of DJ K (S ) - using an Adams-Hilton model;

specify Hurewicz images

- using a minimal Quillen model;

use naturality to extend the calculations to the loop homology of DJ _K ;

calculate the loop homology of Z _K - Lie algebra calculation; geometrical realisation.

(48)

Strategy

calculate the rational loop homology of DJ K (S ) - using an Adams-Hilton model;

specify Hurewicz images

- using a minimal Quillen model;

use naturality to extend the calculations to the loop homology of DJ _K ;

calculate the loop homology of Z _K

- Lie algebra calculation;

geometrical realisation.

(49)

Strategy

calculate the rational loop homology of DJ K (S ) - using an Adams-Hilton model;

specify Hurewicz images

- using a minimal Quillen model;

use naturality to extend the calculations to the loop homology of DJ _K ;

calculate the loop homology of Z _K - Lie algebra calculation; geometrical realisation.

(50)

Strategy

calculate the rational loop homology of DJ K (S ) - using an Adams-Hilton model;

specify Hurewicz images

- using a minimal Quillen model;

use naturality to extend the calculations to the loop homology of DJ _K ;

calculate the loop homology of Z _K - Lie algebra calculation;

geometrical realisation.

(51)

Strategy

calculate the rational loop homology of DJ K (S ) - using an Adams-Hilton model;

specify Hurewicz images

- using a minimal Quillen model;

use naturality to extend the calculations to the loop homology of DJ _K ;

calculate the loop homology of Z _K - Lie algebra calculation;

geometrical realisation.

(52)

Properties of ΩDJ _K and ΩZ _K for MF -complexes

There is a homotopy fibration diagram Z _K (S ² ) ^//

Z

_K

(ı)

DJ K (S ² ) ^//

DJ

K

(ı)

Q r i=1 S ²

Q

r i=1

ı

Z _K ^// DJ _K ^// Q r

i=1 C P ^∞ .

Theorem

Let K be an MF -complex. There is an algebra iso H ∗ (ΩDJ _K ; Q ) ∼ = U(L _ab hb ₁ , . . . , b n i `

Lhu _σ | σ ∈ MF (K )i)/(I + J)

where u σ is the Hurewicz image of a higher Samelson product,

I = (b ² _i , [u σ , b _j

_σ

] | 1 ≤ i ≤ n, σ = (i 1 , . . . , i _k ) ∈ MF (K ), j σ ∈ σ).

(53)

Properties of ΩDJ _K and ΩZ _K for MF -complexes

There is a homotopy fibration diagram Z _K (S ² ) ^//

Z

_K

(ı)

DJ K (S ² ) ^//

DJ

K

(ı)

Q r i=1 S ²

Q

r i=1

ı

Z _K ^// DJ _K ^// Q r

i=1 C P ^∞ . Theorem

Let K be an MF -complex. There is an algebra iso H ∗ (ΩDJ _K ; Q ) ∼ = U(L _ab hb ₁ , . . . , b n i `

Lhu _σ | σ ∈ MF(K )i)/(I + J) where u σ is the Hurewicz image of a higher Samelson product, I = (b ² _i , [u _σ , b _j

_σ

] | 1 ≤ i ≤ n, σ = (i ₁ , . . . , i _k ) ∈ MF (K ), j _σ ∈ σ).

(54)

Proposition

There is a commutative diagram of algebras

H ∗ (ΩZ _K ; Q) ^//

∼ =

H ∗ (ΩDJ K ; Q)

∼ =

ULh Ri e ^U(˜ ⁱ ⁾ ^// U (L _ab hb ₁ , . . . , b n i `

Lhu _σ | σ ∈ MF (K )i)/I

where R e = {[[u _σ , b _j

₁

], . . . , b _j

_l

] | σ ∈ MF (K ), {j ₁ , . . . , j _l } ⊆ [n]\σ,

1 ≤ j 1 < · · · < j l ≤ n, 0 ≤ l ≤ n}.

(55)

Geometrical realisation

Theorem

Let K be an MF -complex on n vertices, The map _

α∈e e I

S ^t

^α^e

−→ Z ^' _K −→ DJ _K

is a wedge sum of the following maps:

(a) a higher Whitehead product w e σ : S ^2|σ|+1 −→ DJ K for each missing face σ ∈ MF (K );

(b) an iterated Whitehead product

[[ w e σ , ˜ a j

1

] . . . , ˜ a j

l

] : S ^2|σ|+l+1 −→ DJ K for each σ ∈ MF (K ) of dimension greater than 1 and each list j 1 < · · · < j _l in [n]\σ, where 1 ≤ l ≤ n;

(c) the collection of independent iterated Whitehead products W f σ

for each σ ∈ MF(K ) of dimension 1.

(56)

Example

K =

(1), (2), (3), (4), (1, 2), (1, 3), (1, 4), (2, 3), (2, 4) .

MF(K ) =

(3, 4), (1, 2, 3), (1, 2, 4) . H ∗ (ΩDJ _K (S )) ∼ = U (L _ab hb ₁ , b 2 , b 3 , b 4 i `

Lhu ₁ , u 2 , u 3 i) /J where u ₁ , u ₂ , u ₃ are the Hurewicz images.

Jacobi identity:

[u 1 , b 1 ] = [[b 3 , b 4 ], b 1 ] = [b 3 , [b 4 , b 1 ]] + [b 4 , [b 3 , b 1 ]]. Therefore [u ₁ , b ₁ ] = 0 and [u ₁ , b ₂ ] = 0. Thus J = ([u 1 , b 1 ], [u 1 , b 2 ]).

H ∗ (ΩDJ _K ) ∼ = U (L _ab hb ₁ , b 2 , b 3 , b 4 i `

Lhu ₁ , u 2 , u 3 i) /(I + J) u ₁ ⇐⇒ w e ₁ : S ³ −→ C P ₃ ^∞ ∨ C P ₄ ^∞ −→ DJ _K

u 2 ⇐⇒ w e 2 : S ⁵ −→ FW (1, 2, 3) −→ DJ K

u 3 ⇐⇒ w e 3 : S ⁵ −→ FW (1, 2, 4) −→ DJ _K I =

(b _i ² , [u 1 , b 3 ], [u 1 , b 4 ], [u 2 , b 1 ], [u 2 , b 2 ], [u 2 , b 3 ], [u 3 , b 1 ], [u 3 , b 2 ], [u 3 , b 4 ]).

(57)

Example

K =

(1), (2), (3), (4), (1, 2), (1, 3), (1, 4), (2, 3), (2, 4) . MF(K ) =

(3, 4), (1, 2, 3), (1, 2, 4) .

H ∗ (ΩDJ _K (S )) ∼ = U (L _ab hb ₁ , b 2 , b 3 , b 4 i `

Lhu ₁ , u 2 , u 3 i) /J where u ₁ , u ₂ , u ₃ are the Hurewicz images.

Jacobi identity:

[u 1 , b 1 ] = [[b 3 , b 4 ], b 1 ] = [b 3 , [b 4 , b 1 ]] + [b 4 , [b 3 , b 1 ]]. Therefore [u ₁ , b ₁ ] = 0 and [u ₁ , b ₂ ] = 0. Thus J = ([u 1 , b 1 ], [u 1 , b 2 ]).

H ∗ (ΩDJ _K ) ∼ = U (L _ab hb ₁ , b 2 , b 3 , b 4 i `

Lhu ₁ , u 2 , u 3 i) /(I + J) u ₁ ⇐⇒ w e ₁ : S ³ −→ C P ₃ ^∞ ∨ C P ₄ ^∞ −→ DJ _K

u 2 ⇐⇒ w e 2 : S ⁵ −→ FW (1, 2, 3) −→ DJ K

u 3 ⇐⇒ w e 3 : S ⁵ −→ FW (1, 2, 4) −→ DJ _K I =

(b _i ² , [u 1 , b 3 ], [u 1 , b 4 ], [u 2 , b 1 ], [u 2 , b 2 ], [u 2 , b 3 ], [u 3 , b 1 ], [u 3 , b 2 ], [u 3 , b 4 ]).

(58)

Example

K =

(1), (2), (3), (4), (1, 2), (1, 3), (1, 4), (2, 3), (2, 4) . MF(K ) =

(3, 4), (1, 2, 3), (1, 2, 4) . H ∗ (ΩDJ _K (S )) ∼ = U (L _ab hb ₁ , b ₂ , b ₃ , b ₄ i `

Lhu ₁ , u ₂ , u ₃ i) /J where u ₁ , u ₂ , u ₃ are the Hurewicz images.

Jacobi identity:

[u 1 , b 1 ] = [[b 3 , b 4 ], b 1 ] = [b 3 , [b 4 , b 1 ]] + [b 4 , [b 3 , b 1 ]]. Therefore [u ₁ , b ₁ ] = 0 and [u ₁ , b ₂ ] = 0. Thus J = ([u 1 , b 1 ], [u 1 , b 2 ]).

H ∗ (ΩDJ _K ) ∼ = U (L _ab hb ₁ , b 2 , b 3 , b 4 i `

Lhu ₁ , u 2 , u 3 i) /(I + J) u ₁ ⇐⇒ w e ₁ : S ³ −→ C P ₃ ^∞ ∨ C P ₄ ^∞ −→ DJ _K

u 2 ⇐⇒ w e 2 : S ⁵ −→ FW (1, 2, 3) −→ DJ K

u 3 ⇐⇒ w e 3 : S ⁵ −→ FW (1, 2, 4) −→ DJ _K I =

(b _i ² , [u 1 , b 3 ], [u 1 , b 4 ], [u 2 , b 1 ], [u 2 , b 2 ], [u 2 , b 3 ], [u 3 , b 1 ], [u 3 , b 2 ], [u 3 , b 4 ]).

(59)

Example

K =

(1), (2), (3), (4), (1, 2), (1, 3), (1, 4), (2, 3), (2, 4) . MF(K ) =

(3, 4), (1, 2, 3), (1, 2, 4) . H ∗ (ΩDJ _K (S )) ∼ = U (L _ab hb ₁ , b ₂ , b ₃ , b ₄ i `

Lhu ₁ , u ₂ , u ₃ i) /J where u ₁ , u ₂ , u ₃ are the Hurewicz images.

Jacobi identity:

[u 1 , b 1 ] = [[b 3 , b 4 ], b 1 ] = [b 3 , [b 4 , b 1 ]] + [b 4 , [b 3 , b 1 ]].

Therefore [u ₁ , b ₁ ] = 0 and [u ₁ , b ₂ ] = 0. Thus J = ([u 1 , b 1 ], [u 1 , b 2 ]).

H ∗ (ΩDJ _K ) ∼ = U (L _ab hb ₁ , b 2 , b 3 , b 4 i `

Lhu ₁ , u 2 , u 3 i) /(I + J) u ₁ ⇐⇒ w e ₁ : S ³ −→ C P ₃ ^∞ ∨ C P ₄ ^∞ −→ DJ _K

u 2 ⇐⇒ w e 2 : S ⁵ −→ FW (1, 2, 3) −→ DJ K

u 3 ⇐⇒ w e 3 : S ⁵ −→ FW (1, 2, 4) −→ DJ _K I =

(b _i ² , [u 1 , b 3 ], [u 1 , b 4 ], [u 2 , b 1 ], [u 2 , b 2 ], [u 2 , b 3 ], [u 3 , b 1 ], [u 3 , b 2 ], [u 3 , b 4 ]).

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Example

K =

(1), (2), (3), (4), (1, 2), (1, 3), (1, 4), (2, 3), (2, 4) . MF(K ) =

(3, 4), (1, 2, 3), (1, 2, 4) . H ∗ (ΩDJ _K (S )) ∼ = U (L _ab hb ₁ , b ₂ , b ₃ , b ₄ i `

Lhu ₁ , u ₂ , u ₃ i) /J where u ₁ , u ₂ , u ₃ are the Hurewicz images.

Jacobi identity:

[u 1 , b 1 ] = [[b 3 , b 4 ], b 1 ] = [b 3 , [b 4 , b 1 ]] + [b 4 , [b 3 , b 1 ]].

Therefore [u ₁ , b ₁ ] = 0 and [u ₁ , b ₂ ] = 0. Thus J = ([u 1 , b 1 ], [u 1 , b 2 ]).

H ∗ (ΩDJ _K ) ∼ = U (L _ab hb ₁ , b 2 , b 3 , b 4 i `

Lhu ₁ , u 2 , u 3 i) /(I + J) u ₁ ⇐⇒ w e ₁ : S ³ −→ C P ₃ ^∞ ∨ C P ₄ ^∞ −→ DJ _K

u 2 ⇐⇒ w e 2 : S ⁵ −→ FW (1, 2, 3) −→ DJ K

u 3 ⇐⇒ w e 3 : S ⁵ −→ FW (1, 2, 4) −→ DJ _K I =

(b _i ² , [u 1 , b 3 ], [u 1 , b 4 ], [u 2 , b 1 ], [u 2 , b 2 ], [u 2 , b 3 ], [u 3 , b 1 ], [u 3 , b 2 ], [u 3 , b 4 ]).

(61)

The wedge summands of Z _K and the maps to DJ K are as follows.

Part (a): S ³ , S ⁵ and S ⁵ with maps w e 1 , w e 2 and w e 3

Part (b): S ⁶ and S ⁶ from iterated Whitehead products [ w e 2 , ˜ a 4 ] : S ⁶ −→ DJ _K and [ w e 3 , ˜ a 3 ] : S ⁶ −→ DJ _K . Part (c) is vacuous in this case.

S ³ ∨ 2S ⁵ ∨ 2S ⁶ −→ DJ _K

which is the wedge sum of w e ₁ , w e ₂ , w e ₃ , [ w e ₂ , ˜ a ₄ ] and [ w e ₃ , ˜ a ₃ ].

(62)

The wedge summands of Z _K and the maps to DJ K are as follows.

Part (a): S ³ , S ⁵ and S ⁵ with maps w e 1 , w e 2 and w e 3

Part (b): S ⁶ and S ⁶ from iterated Whitehead products [ w e 2 , ˜ a 4 ] : S ⁶ −→ DJ _K and [ w e 3 , ˜ a 3 ] : S ⁶ −→ DJ _K .

Part (c) is vacuous in this case.

S ³ ∨ 2S ⁵ ∨ 2S ⁶ −→ DJ _K

which is the wedge sum of w e ₁ , w e ₂ , w e ₃ , [ w e ₂ , ˜ a ₄ ] and [ w e ₃ , ˜ a ₃ ].

(63)

The wedge summands of Z _K and the maps to DJ K are as follows.

Part (a): S ³ , S ⁵ and S ⁵ with maps w e 1 , w e 2 and w e 3

Part (b): S ⁶ and S ⁶ from iterated Whitehead products [ w e 2 , ˜ a 4 ] : S ⁶ −→ DJ _K and [ w e 3 , ˜ a 3 ] : S ⁶ −→ DJ _K . Part (c) is vacuous in this case.

S ³ ∨ 2S ⁵ ∨ 2S ⁶ −→ DJ _K

which is the wedge sum of w e ₁ , w e ₂ , w e ₃ , [ w e ₂ , ˜ a ₄ ] and [ w e ₃ , ˜ a ₃ ].

(64)

The wedge summands of Z _K and the maps to DJ K are as follows.

Part (a): S ³ , S ⁵ and S ⁵ with maps w e 1 , w e 2 and w e 3

Part (b): S ⁶ and S ⁶ from iterated Whitehead products [ w e 2 , ˜ a 4 ] : S ⁶ −→ DJ _K and [ w e 3 , ˜ a 3 ] : S ⁶ −→ DJ _K . Part (c) is vacuous in this case.

S ³ ∨ 2S ⁵ ∨ 2S ⁶ −→ DJ _K

which is the wedge sum of w e ₁ , w e ₂ , w e ₃ , [ w e ₂ , ˜ a ₄ ] and [ w e ₃ , ˜ a ₃ ].

(65)

Contributions to Algebra

Problem: The nature of Tor _k[K] (k, k).

The Poincar´ e series P (k [K ]) =

∞

X

i=0

b _i t ⁱ where b _i = dim _k Tor ^k[K] _i (k, k )

Problem: The rationality of P (k [K ]). Theorem

(Golod) There exist non-negative integers n, c 1 , . . . , c n such that P (k[K ]) ≤ (1 + t) ⁿ

1 − P n

i=1 c i t ⁱ⁺¹ . Theorem

(G.,Theriault) There is a topological proof of Golod’s inequality.

(66)

Contributions to Algebra

Problem: The nature of Tor _k[K] (k, k).

The Poincar´ e series P (k [K ]) =

∞

X

i=0

b _i t ⁱ where b _i = dim _k Tor ^k[K] _i (k, k ) Problem: The rationality of P (k [K ]).

Theorem

(Golod) There exist non-negative integers n, c ₁ , . . . , c _n such that P (k[K ]) ≤ (1 + t) ⁿ

1 − P n

i=1 c i t ⁱ⁺¹ .

Theorem

(G.,Theriault) There is a topological proof of Golod’s inequality.

(67)

Contributions to Algebra

Problem: The nature of Tor _k[K] (k, k).

The Poincar´ e series P (k [K ]) =

∞

X

i=0

b _i t ⁱ where b _i = dim _k Tor ^k[K] _i (k, k ) Problem: The rationality of P (k [K ]).

Theorem

(Golod) There exist non-negative integers n, c ₁ , . . . , c _n such that P (k[K ]) ≤ (1 + t) ⁿ

1 − P n

i=1 c i t ⁱ⁺¹ . Theorem

(G.,Theriault) There is a topological proof of Golod’s inequality.

(68)

Theorem

Tor ^∗ _k[K] (k , k) ∼ = H ^∗ (ΩDJ(K ); k).

Looking at the split fibration ΩZ _K −→ ΩDJ(K ) −→ T ⁿ Tor ^∗ _k[K _] (k, k ) ∼ = H ^∗ (ΩDJ(K )) = H ^∗ (T ⁿ ) ⊗ H ^∗ (ΩZ _K ) Using the bar resolution,

P(H ^∗ (ΩZ _K )) ≤ P (T (Σ ⁻¹ H ^∗ (Z _K ))). Therefore

P (R) = (1 + t) ⁿ P (H ^∗ (ΩZ _K )) ≤ (1 + t) ⁿ P (T (Σ ⁻¹ H ^∗ (Z _K )))

= (1 + t ) ⁿ 1 − P (Σ ⁻¹ H ^∗ (Z _K )) . Equality is obtained when H ^∗ (Z _K ) is Golod.

Corollary

When K in our family, then P(k[K ]) is rational.

(69)

Theorem

Tor ^∗ _k[K] (k , k) ∼ = H ^∗ (ΩDJ(K ); k).

Looking at the split fibration ΩZ _K −→ ΩDJ(K ) −→ T ⁿ Tor ^∗ _k[K _] (k , k) ∼ = H ^∗ (ΩDJ(K )) = H ^∗ (T ⁿ ) ⊗ H ^∗ (ΩZ _K )

Using the bar resolution,

P(H ^∗ (ΩZ _K )) ≤ P (T (Σ ⁻¹ H ^∗ (Z _K ))). Therefore

P (R) = (1 + t) ⁿ P (H ^∗ (ΩZ _K )) ≤ (1 + t) ⁿ P (T (Σ ⁻¹ H ^∗ (Z _K )))

= (1 + t ) ⁿ 1 − P (Σ ⁻¹ H ^∗ (Z _K )) . Equality is obtained when H ^∗ (Z _K ) is Golod.

Corollary

When K in our family, then P(k[K ]) is rational.

(70)

Theorem

Tor ^∗ _k[K] (k , k) ∼ = H ^∗ (ΩDJ(K ); k).

Looking at the split fibration ΩZ _K −→ ΩDJ(K ) −→ T ⁿ Tor ^∗ _k[K _] (k , k) ∼ = H ^∗ (ΩDJ(K )) = H ^∗ (T ⁿ ) ⊗ H ^∗ (ΩZ _K ) Using the bar resolution,

P (H ^∗ (ΩZ _K )) ≤ P (T (Σ ⁻¹ H ^∗ (Z _K ))).

Therefore

P (R) = (1 + t) ⁿ P (H ^∗ (ΩZ _K )) ≤ (1 + t) ⁿ P (T (Σ ⁻¹ H ^∗ (Z _K )))

= (1 + t ) ⁿ 1 − P (Σ ⁻¹ H ^∗ (Z _K )) . Equality is obtained when H ^∗ (Z _K ) is Golod.

Corollary

When K in our family, then P(k[K ]) is rational.

(71)

Theorem

Tor ^∗ _k[K] (k , k) ∼ = H ^∗ (ΩDJ(K ); k).

Looking at the split fibration ΩZ _K −→ ΩDJ(K ) −→ T ⁿ Tor ^∗ _k[K _] (k , k) ∼ = H ^∗ (ΩDJ(K )) = H ^∗ (T ⁿ ) ⊗ H ^∗ (ΩZ _K ) Using the bar resolution,

P (H ^∗ (ΩZ _K )) ≤ P (T (Σ ⁻¹ H ^∗ (Z _K ))).

Therefore

P (R) = (1 + t) ⁿ P (H ^∗ (ΩZ _K )) ≤ (1 + t) ⁿ P (T (Σ ⁻¹ H ^∗ (Z _K )))

= (1 + t) ⁿ 1 − P (Σ ⁻¹ H ^∗ (Z _K )) . Equality is obtained when H ^∗ (Z _K ) is Golod.

Corollary

When K in our family, then P(k [K ]) is rational.

25th Nordic and 1st British-Nordic congress of Mathematicians