Relative L-S category and categorical length

Relative L-S category and categorical length

Norio IWASE

Faculty of Mathematics, Kyushu University

ALGEBRAIC TOPOLOGY: OLD AND NEW M.M.POSTNIKOV MEMORIAL CONFERENCE,

Bedlewo, Poland,18-24 June 2007

L-S category

Definition (Lusternik-Schnirelmann)









 m≥0

∃{A₀, ...,A_m; closed in M}

L-S category

Definition (Lusternik-Schnirelmann)









 m≥0

∃{A₀, ...,A_m; open in M}

M=S_m

i=0A_i, where each Ai is contractibleinM.











L-S category

Definition (Lusternik-Schnirelmann)









 m≥0

∃{A₀, ...,A_m; in M}

M=S_m

i=0A_i, where each Ai is contractibleinM.











L-S category

Definition (Lusternik-Schnirelmann)









 m≥0

∃{A₀, ...,A_m; in M}

M=S_m

i=0A_i, where each Ai is contractibleinM.











Remark:

L-S category

Definition (Lusternik-Schnirelmann)









 m≥0

∃{A₀, ...,A_m; in M}

M=S_m

i=0A_i, where each Ai is contractibleinM.











Remark:

We don’t have any means

to know how good is the given covering.

L-S category

Definition (Lusternik-Schnirelmann)









 m≥0

∃{A₀, ...,A_m; in M}

M=S_m

i=0A_i, where each Ai is contractibleinM.











Remark:

We don’t have any means

to know how good is the given covering.

That is why,

L-S category

Definition (Lusternik-Schnirelmann)









 m≥0

∃{A₀, ...,A_m; in M}

M=S_m

i=0A_i, where each Ai is contractibleinM.











Remark:

We don’t have any means

to know how good is the given covering.

That is why, this definition

gives only anupper boundforcat(M).

L-S category

G. Whitehead gave an alternative definition of L-S category.

L-S category

G. Whitehead gave an alternative definition of L-S category.

Definition (Whitehead)

(

m≥0 ∆^m+1: M→Q_m+1

M is compressible into thefat wedgeT^m+1M⊆Q_m+1

M.

)

L-S category

G. Whitehead gave an alternative definition of L-S category.

Definition (Whitehead)

(

m≥0 ∆^m+1: M→Q_m+1

M is compressible into thefat wedgeT^m+1M⊆Q_m+1

M.

)

whereT^m+1M={(x0, ...,xm) ∃_ixi=∗}.

L-S category

G. Whitehead gave an alternative definition of L-S category.

Definition (Whitehead)

(

m≥0 ∆^m+1: M→Q_m+1

M is compressible into thefat wedgeT^m+1M⊆Q_m+1

M.

)

whereT^m+1M={(x0, ...,xm) ∃_ixi=∗}.

The Whitehead’s definition coincides with the original one for a good space.

L-S category

G. Whitehead gave an alternative definition of L-S category.

Definition (Whitehead)

(

m≥0 ∆^m+1: M→Q_m+1

M is compressible into thefat wedgeT^m+1M⊆Q_m+1

M.

)

whereT^m+1M={(x0, ...,xm) ∃_ixi=∗}.

The Whitehead’s definition coincides with the original one for a good space. Moreover from this definition, we obtain

L-S category

G. Whitehead gave an alternative definition of L-S category.

Definition (Whitehead)

(

m≥0 ∆^m+1: M→Q_m+1

M is compressible into thefat wedgeT^m+1M⊆Q_m+1

M.

)

whereT^m+1M={(x0, ...,xm) ∃_ixi=∗}.

The Whitehead’s definition coincides with the original one for a good space. Moreover from this definition, we obtain

Theorem (Whitehead)

For any ringR,cat(X)is bounded below by thecup-length

L-S category

G. Whitehead gave an alternative definition of L-S category.

Definition (Whitehead)

(

m≥0 ∆^m+1: M→Q_m+1

M is compressible into thefat wedgeT^m+1M⊆Q_m+1

M.

)

whereT^m+1M={(x0, ...,xm) ∃_ixi=∗}.

The Whitehead’s definition coincides with the original one for a good space. Moreover from this definition, we obtain

Theorem (Whitehead)

For any ringR,cat(X)is bounded below by thecup-length cup(X; R)=Min

m≥0 ∀_{u0,...,um∈H}^∗_(M;R)u0·u1· · ·um=0

Element of Hopf invariant one

Let us recall the following classical result: if ann-sphere is a Hopf space, then there must be a Hopf invariant one element inπ_2n+1(Sⁿ⁺¹). The first non-trivial case, whenn=15was solved in negative by Toda and

Theorem (Adams)

Element of Hopf invariant one exists inπ_2n+1(Sⁿ⁺¹)if and only if n=0,1,3,7.

In other words,

Claim

“Hopf invariants detects Hopf structures.”

Element of Hopf invariant one

Let us recall the following classical result: if ann-sphere is a Hopf space, then there must be a Hopf invariant one element inπ_2n+1(Sⁿ⁺¹). The first non-trivial case, whenn=15was solved in negative by Toda and

Theorem (Adams)

Element of Hopf invariant one exists inπ_2n+1(Sⁿ⁺¹)if and only if n=0,1,3,7.

In other words,

Claim

“Hopf invariants detects Hopf structures.”

Element of Hopf invariant one

Let us recall the following classical result: if ann-sphere is a Hopf space, then there must be a Hopf invariant one element inπ_2n+1(Sⁿ⁺¹). The first non-trivial case, whenn=15was solved in negative by Toda and

Theorem (Adams)

Element of Hopf invariant one exists inπ_2n+1(Sⁿ⁺¹)if and only if n=0,1,3,7.

In other words,

Claim

“Hopf invariants detects Hopf structures.”

Element of Hopf invariant one

Let us recall the following classical result: if ann-sphere is a Hopf space, then there must be a Hopf invariant one element inπ_2n+1(Sⁿ⁺¹). The first non-trivial case, whenn=15was solved in negative by Toda and

Theorem (Adams)

Element of Hopf invariant one exists inπ_2n+1(Sⁿ⁺¹)if and only if n=0,1,3,7.

In other words,

Claim

“Hopf invariants detects Hopf structures.”

Element of Hopf invariant one

Let us recall the following classical result: if ann-sphere is a Hopf space, then there must be a Hopf invariant one element inπ_2n+1(Sⁿ⁺¹). The first non-trivial case, whenn=15was solved in negative by Toda and

Theorem (Adams)

Element of Hopf invariant one exists inπ_2n+1(Sⁿ⁺¹)if and only if n=0,1,3,7.

In other words,

Claim

“Hopf invariants detects Hopf structures.”

Berstein-Hilton’s criterion

Let us consider the following2-cell complexes:

Example

1 RP²,CP²,HP²,OP² — (projective planes)

2 Q2 =S³∪ωe⁷⊂ Sp(2) — James’ quasi projective plane.

3 S¹∪e²⊂L³(p,q) — the2-skeleton of a lens spaceL³(p,q).

4 etc. · · ·

We may write them asX=S^r∪_f e^q+1.

Theorem (Berstein-Hilton)

cat(X)=2 ⇐⇒ H1( f ),0

Berstein-Hilton’s criterion

Let us consider the following2-cell complexes:

Example

1 RP²,CP²,HP²,OP² — (projective planes)

2 Q2 =S³∪ωe⁷⊂ Sp(2) — James’ quasi projective plane.

3 S¹∪e²⊂L³(p,q) — the2-skeleton of a lens spaceL³(p,q).

4 etc. · · ·

We may write them asX=S^r∪_f e^q+1.

Theorem (Berstein-Hilton)

cat(X)=2 ⇐⇒ H1( f ),0

Berstein-Hilton’s criterion

Let us consider the following2-cell complexes:

Example

1 RP²,CP²,HP²,OP² — (projective planes)

2 Q2 =S³∪ωe⁷⊂ Sp(2) — James’ quasi projective plane.

3 S¹∪e²⊂L³(p,q) — the2-skeleton of a lens spaceL³(p,q).

4 etc. · · ·

We may write them asX=S^r∪_f e^q+1.

Theorem (Berstein-Hilton)

cat(X)=2 ⇐⇒ H1( f ),0

Berstein-Hilton’s criterion

Let us consider the following2-cell complexes:

Example

1 RP²,CP²,HP²,OP² — (projective planes)

2 Q2 =S³∪ωe⁷⊂ Sp(2) — James’ quasi projective plane.

3 S¹∪e²⊂L³(p,q) — the2-skeleton of a lens spaceL³(p,q).

4 etc. · · ·

We may write them asX=S^r∪_f e^q+1.

Theorem (Berstein-Hilton)

cat(X)=2 ⇐⇒ H1( f ),0

Berstein-Hilton’s criterion

Let us consider the following2-cell complexes:

Example

1 RP²,CP²,HP²,OP² — (projective planes)

2 Q2 =S³∪ωe⁷⊂ Sp(2) — James’ quasi projective plane.

3 S¹∪e²⊂L³(p,q) — the2-skeleton of a lens spaceL³(p,q).

4 etc. · · ·

We may write them asX=S^r∪_f e^q+1.

Theorem (Berstein-Hilton)

cat(X)=2 ⇐⇒ H1( f ),0

Berstein-Hilton’s criterion

Let us consider the following2-cell complexes:

Example

1 RP²,CP²,HP²,OP² — (projective planes)

2 Q2 =S³∪ωe⁷⊂ Sp(2) — James’ quasi projective plane.

3 S¹∪e²⊂L³(p,q) — the2-skeleton of a lens spaceL³(p,q).

4 etc. · · ·

We may write them asX=S^r∪_f e^q+1.

Theorem (Berstein-Hilton)

cat(X)=2 ⇐⇒ H1( f ),0

Berstein-Hilton’s criterion

Let us consider the following2-cell complexes:

Example

1 RP²,CP²,HP²,OP² — (projective planes)

2 Q2 =S³∪ωe⁷⊂ Sp(2) — James’ quasi projective plane.

3 S¹∪e²⊂L³(p,q) — the2-skeleton of a lens spaceL³(p,q).

4 etc. · · ·

We may write them asX=S^r∪_f e^q+1.

Theorem (Berstein-Hilton)

cat(X)=2 ⇐⇒ H1( f ),0

Berstein-Hilton’s criterion

Let us consider the following2-cell complexes:

Example

1 RP²,CP²,HP²,OP² — (projective planes)

2 Q2 =S³∪ωe⁷⊂ Sp(2) — James’ quasi projective plane.

3 S¹∪e²⊂L³(p,q) — the2-skeleton of a lens spaceL³(p,q).

4 etc. · · ·

We may write them asX=S^r∪_f e^q+1.

Theorem (Berstein-Hilton)

cat(X)=2 ⇐⇒ H₁( f ),0inπ_q+1(S^r×S^r,S^r∨S^r)

Higher Hopf invariants

Definition (Berstein-Hilton)

For a map f fromS^qto a spaceXwithcat(X)=m, H_m^s( f )∈π_q+1Q_m+1

X,T^m+1X ,

where sis a compression of them-fold diagonal∆^m+1 : X→Qm+1Xinto the fat wedgeT^m+1X.

Theorem (Ganea)

X,T^m+1X)π_q(Σ^mV_m+1 Ω(X))

What does it mean?

Higher Hopf invariants

Definition (Berstein-Hilton)

For a map f fromS^qto a spaceXwithcat(X)=m, H_m^s( f )∈π_q+1Q_m+1

X,T^m+1X ,

where sis a compression of them-fold diagonal∆^m+1 : X→Qm+1Xinto the fat wedgeT^m+1X.

Theorem (Ganea)

X,T^m+1X)π_q(Σ^mV_m+1 Ω(X))

What does it mean?

Higher Hopf invariants

Definition (Berstein-Hilton)

For a map f fromS^qto a spaceXwithcat(X)=m, H_m^s( f )∈π_q+1Q_m+1

X,T^m+1X ,

where sis a compression of them-fold diagonal∆^m+1 : X→Qm+1Xinto the fat wedgeT^m+1X.

Theorem (Ganea)

X,T^m+1X)π_q(Σ^mV_m+1 Ω(X))

What does it mean?

Higher Hopf invariants

Definition (Berstein-Hilton)

For a map f fromS^qto a spaceXwithcat(X)=m, H_m^s( f )∈π_q+1Q_m+1

X,T^m+1X ,

where sis a compression of them-fold diagonal∆^m+1 : X→Qm+1Xinto the fat wedgeT^m+1X.

Theorem (Ganea)

X,T^m+1X)π_q(Σ^mV_m+1 Ω(X))

What does it mean?

A_m

-structure on a space

As a Hopf invariant detects a Hopf structure,

Claim

A higher Hopf inavariant shoulddetect anA_m-structure a higher homotopy associativity,

Here we can recall that

1 A₁space is just a space with a base point.

2 A₂space is a Hopf space with strict unit1.

3 A3space is a homotopy associative Hopf space with strict unit1.

4 Amspace is a ‘higher’ homotopy associative Hopf space with strict unit1.

A_m

-structure on a space

As a Hopf invariant detects a Hopf structure,

Claim

A higher Hopf inavariant shoulddetect anA_m-structure a higher homotopy associativity,

Here we can recall that

1 A₁space is just a space with a base point.

2 A₂space is a Hopf space with strict unit1.

3 A3space is a homotopy associative Hopf space with strict unit1.

4 Amspace is a ‘higher’ homotopy associative Hopf space with strict unit1.

A_m

-structure on a space

As a Hopf invariant detects a Hopf structure,

Claim

A higher Hopf inavariant shoulddetect anA_m-structure a higher homotopy associativity,

Here we can recall that

1 A₁space is just a space with a base point.

2 A₂space is a Hopf space with strict unit1.

3 A3space is a homotopy associative Hopf space with strict unit1.

4 Amspace is a ‘higher’ homotopy associative Hopf space with strict unit1.

A_m

-structure on a space

As a Hopf invariant detects a Hopf structure,

Claim

A higher Hopf inavariant shoulddetect anA_m-structure a higher homotopy associativity,

Here we can recall that

1 A₁space is just a space with a base point.

2 A₂space is a Hopf space with strict unit1.

3 A3space is a homotopy associative Hopf space with strict unit1.

4 Amspace is a ‘higher’ homotopy associative Hopf space with strict unit1.

A_m

-structure on a space

As a Hopf invariant detects a Hopf structure,

Claim

A higher Hopf inavariant shoulddetect anA_m-structure a higher homotopy associativity,

Here we can recall that

1 A₁space is just a space with a base point.

2 A₂space is a Hopf space with strict unit1.

3 A3space is a homotopy associative Hopf space with strict unit1.

4 Amspace is a ‘higher’ homotopy associative Hopf space with strict unit1.

A_m

-structure on a space

As a Hopf invariant detects a Hopf structure,

Claim

A higher Hopf inavariant shoulddetect anA_m-structure a higher homotopy associativity,

Here we can recall that

1 A₁space is just a space with a base point.

2 A₂space is a Hopf space with strict unit1.

3 A3space is a homotopy associative Hopf space with strict unit1.

4 Amspace is a ‘higher’ homotopy associative Hopf space with strict unit1.

A_m

-structure on a space

As a Hopf invariant detects a Hopf structure,

Claim

A higher Hopf inavariant shoulddetect anA_m-structure a higher homotopy associativity,

Here we can recall that

1 A₁space is just a space with a base point.

2 A₂space is a Hopf space with strict unit1.

3 A3space is a homotopy associative Hopf space with strict unit1.

4 Amspace is a ‘higher’ homotopy associative Hopf space with strict unit1.

A∞

-structure on a space

Theorem (Stasheff)

For anyX, the (based) loop spaceΩ(X)ofXadmits a naturalA∞-structure, a sequence of fibrations over projective spacesP^mΩ(X)with fibreΩ(X),

p^Ω(X)_m : E^m+1Ω(X)→P^mΩ(X), E^m+1Ω(X)≃Σ^mV_m+1 Ω(X), and natural classifying mapse_m^X : P^mΩ(X)−→X.

Theorem (Ganea,I,Sakai)

cat(X)=m ⇐⇒ ∃σ: X →P^mΩ(X)such thate^X_mσ∼1X.

One of the advantage to consider projective spaces is that a higher Hopf invariant can be defined on the well-studied projective spaces.

A∞

-structure on a space

Theorem (Stasheff)

For anyX, the (based) loop spaceΩ(X)ofXadmits a naturalA∞-structure, a sequence of fibrations over projective spacesP^mΩ(X)with fibreΩ(X),

p^Ω(X)_m : E^m+1Ω(X)→P^mΩ(X), E^m+1Ω(X)≃Σ^mV_m+1 Ω(X), and natural classifying mapse_m^X : P^mΩ(X)−→X.

Theorem (Ganea,I,Sakai)

cat(X)=m ⇐⇒ ∃σ: X →P^mΩ(X)such thate^X_mσ∼1X. One of the advantage to consider projective spaces is that a higher Hopf invariant can be defined on the well-studied projective spaces.

A∞

-structure on a space

Theorem (Stasheff)

For anyX, the (based) loop spaceΩ(X)ofXadmits a naturalA∞-structure, a sequence of fibrations over projective spacesP^mΩ(X)with fibreΩ(X),

p^Ω(X)_m : E^m+1Ω(X)→P^mΩ(X), E^m+1Ω(X)≃Σ^mV_m+1 Ω(X), and natural classifying mapse_m^X : P^mΩ(X)−→X.

Theorem (Ganea,I,Sakai)

cat(X)=m ⇐⇒ ∃σ: X →P^mΩ(X)such thate^X_mσ∼1X. One of the advantage to consider projective spaces is that a higher Hopf invariant can be defined on the well-studied projective spaces.

A∞

-structure on a space

Theorem (Stasheff)

For anyX, the (based) loop spaceΩ(X)ofXadmits a naturalA∞-structure, a sequence of fibrations over projective spacesP^mΩ(X)with fibreΩ(X),

p^Ω(X)_m : E^m+1Ω(X)→P^mΩ(X), E^m+1Ω(X)≃Σ^mV_m+1 Ω(X), and natural classifying mapse_m^X : P^mΩ(X)−→X.

Theorem (Ganea,I,Sakai)

cat(X)=m ⇐⇒ ∃σ: X →P^mΩ(X)such thate^X_mσ∼1X. One of the advantage to consider projective spaces is that a higher Hopf invariant can be defined on the well-studied projective spaces.

A∞

-structure on a space

Theorem (Stasheff)

For anyX, the (based) loop spaceΩ(X)ofXadmits a naturalA∞-structure, a sequence of fibrations over projective spacesP^mΩ(X)with fibreΩ(X),

p^Ω(X)_m : E^m+1Ω(X)→P^mΩ(X), E^m+1Ω(X)≃Σ^mV_m+1 Ω(X), and natural classifying mapse_m^X : P^mΩ(X)−→X.

Theorem (Ganea,I,Sakai)

cat(X)=m ⇐⇒ ∃σ: X →P^mΩ(X)such thate^X_mσ∼1X. One of the advantage to consider projective spaces is that a higher Hopf invariant can be defined on the well-studied projective spaces.

A∞

-structure on a space

Theorem (Stasheff)

For anyX, the (based) loop spaceΩ(X)ofXadmits a naturalA∞-structure, a sequence of fibrations over projective spacesP^mΩ(X)with fibreΩ(X),

p^Ω(X)_m : E^m+1Ω(X)→P^mΩ(X), E^m+1Ω(X)≃Σ^mV_m+1 Ω(X), and natural classifying mapse_m^X : P^mΩ(X)−→X.

Theorem (Ganea,I,Sakai)

cat(X)=m ⇐⇒ ∃σ: X →P^mΩ(X)such thate^X_mσ∼1X. One of the advantage to consider projective spaces is that a higher Hopf invariant can be defined on the well-studied projective spaces.

Projective spaces and a higher Hopf invariant

Claim

LetXbe a space ofcat(X)=m≥1. Then for any f :ΣV→X, we have

Moreover, we can show the following theorem using this definition.

Theorem (I)

If anA_m-structure of anA_m-spaceS can be extended to anA_m+1-structure, then there is an element[ f ]∈[E^m+1(S ),P^m(S )]of higher Hopf invariant one, whereE^m+1(S )= Σ^mV_m+1

S.

Projective spaces and a higher Hopf invariant

Claim

LetXbe a space ofcat(X)=m≥1. Then for any f :ΣV→X, we have H_m^s( f )∈[C(ΣV),ΣV;Q_m+1

X,T^m+1X]

Moreover, we can show the following theorem using this definition.

Theorem (I)

If anAm-structure of anAm-spaceS can be extended to anA_m+1-structure, then there is an element[ f ]∈[E^m+1(S ),P^m(S )]of higher Hopf invariant one, whereE^m+1(S )= Σ^mV_m+1

S.

Projective spaces and a higher Hopf invariant

Claim

LetXbe a space ofcat(X)=m≥1. Then for any f :ΣV→X, we have H_m^s( f )∈[C(ΣV),ΣV;Q_m+1

X,T^m+1X][ΣV,E^m+1Ω(X)]

Moreover, we can show the following theorem using this definition.

Theorem (I)

If anAm-structure of anAm-spaceS can be extended to anA_m+1-structure, then there is an element[ f ]∈[E^m+1(S ),P^m(S )]of higher Hopf invariant one, whereE^m+1(S )= Σ^mV_m+1

S.

Projective spaces and a higher Hopf invariant

Claim

LetXbe a space ofcat(X)=m≥1. Then for any f :ΣV→X, we have Hm( f )⊂[ΣV,E^m+1Ω(X)]kern

(e^X_m)∗: [ΣV,P^mΩ(X)]→[ΣV,X]o . Moreover, we can show the following

theorem using this definition.

Theorem (I)

If anAm-structure of anAm-spaceS can be extended to anA_m+1-structure, then there is an element[ f ]∈[E^m+1(S ),P^m(S )]of higher Hopf invariant one, whereE^m+1(S )= Σ^mV_m+1

S.

Projective spaces and a higher Hopf invariant

Claim

LetXbe a space ofcat(X)=m≥1. Then for any f :ΣV→X, we have Hm( f )⊂[ΣV,E^m+1Ω(X)]kern

(e^X_m)∗: [ΣV,P^mΩ(X)]→[ΣV,X]o . Moreover, we can show the following

theorem using this definition.

Theorem (I)

If anAm-structure of anAm-spaceS can be extended to anA_m+1-structure, then there is an element[ f ]∈[E^m+1(S ),P^m(S )]of higher Hopf invariant one, whereE^m+1(S )= Σ^mV_m+1

S.

Projective spaces and a higher Hopf invariant

Claim

LetXbe a space ofcat(X)=m≥1. Then for any f :ΣV→X, we have Hm( f )⊂[ΣV,E^m+1Ω(X)]kern

(e^X_m)∗: [ΣV,P^mΩ(X)]→[ΣV,X]o . Moreover, we can show the following

theorem using this definition.

Theorem (I)

If anAm-structure of anAm-spaceS can be extended to anA_m+1-structure, then there is an element[ f ]∈[E^m+1(S ),P^m(S )]of higher Hopf invariant one, whereE^m+1(S )= Σ^mV_m+1

S.

Projective spaces and a higher Hopf invariant

Claim

LetXbe a space ofcat(X)=m≥1. Then for any f :ΣV→X, we have Hm( f )⊂[ΣV,E^m+1Ω(X)]kern

(e^X_m)∗: [ΣV,P^mΩ(X)]→[ΣV,X]o . Moreover, we can show the following

theorem using this definition.

Theorem (I)

If anAm-structure of anAm-spaceS can be extended to anA_m+1-structure, then there is an element[ f ]∈[E^m+1(S ),P^m(S )]of higher Hopf invariant one, whereE^m+1(S )= Σ^mV_m+1

S.

Definition (Rudyak, Strom)

wgt(u)=Min (

m≥0 ∃{A0, ...,Am; closed in M}s.t.

M=Sm

i=0Ai, & u|Ai =0∈H^∗(Ai) )

=Min

m≥0 ∃f : A→ Ms.t. cat(A)=m & f^∗(u),0

which can be characterised by using projective spaces:

Proposition

wgt(u)=Minn

m≥0 (e^X_m)^∗(u),0o

Definition (Rudyak, Strom)

wgt(u)=Min (

m≥0 ∃{A0, ...,Am; closed in M}s.t.

M=Sm

i=0Ai, & u|Ai =0∈H^∗(Ai) )

=Min

m≥0 ∃f : A→ Ms.t. cat(A)=m & f^∗(u),0

which can be characterised by using projective spaces:

Proposition

wgt(u)=Minn

m≥0 (e^X_m)^∗(u),0o

Definition (Rudyak, Strom)

wgt(u)=Min (

m≥0 ∃{A0, ...,Am; closed in M}s.t.

M=Sm

i=0Ai, & u|Ai =0∈H^∗(Ai) )

=Min

m≥0 ∃f : A→ Ms.t. cat(A)=m & f^∗(u),0 which can be characterised by using projective spaces:

Proposition

wgt(u)=Minn

m≥0 (e^X_m)^∗(u),0o

Definition (Rudyak, Strom)

wgt(u)=Min (

m≥0 ∃{A0, ...,Am; closed in M}s.t.

M=Sm

i=0Ai, & u|Ai =0∈H^∗(Ai) )

=Min

m≥0 ∃f : A→ Ms.t. cat(A)=m & f^∗(u),0 which can be characterised by using projective spaces:

Proposition

wgt(u)=Minn

m≥0 (e^X_m)^∗(u),0o

Definition (Rudyak, Strom)

wgt(u)=Min (

m≥0 ∃{A0, ...,Am; closed in M}s.t.

M=Sm

i=0Ai, & u|Ai =0∈H^∗(Ai) )

=Min

m≥0 ∃f : A→ Ms.t. cat(A)=m & f^∗(u),0 which can be characterised by using projective spaces:

Proposition

wgt(u)=Minn

m≥0 (e^X_m)^∗(u),0o

Leth^∗be a generalised cohomology theory andh^∗hbe the set of all (unstable) cohomology operations onh^∗.

Definition (I-Kono)

m≥0 (e_m^X)^∗is a split mono ofh^∗h-moduleso

For example, a computation of module weight yields the following.

Theorem (I-Kono)

Mwgt(Spin(9);F₂)≥8whilewgt(Spin(9);F₂)=6.

Leth^∗be a generalised cohomology theory andh^∗hbe the set of all (unstable) cohomology operations onh^∗.

Definition (I-Kono)

m≥0 (e_m^X)^∗is a split mono ofh^∗h-moduleso For example, a computation of module weight yields the following.

Theorem (I-Kono)

Mwgt(Spin(9);F₂)≥8whilewgt(Spin(9);F₂)=6.

Leth^∗be a generalised cohomology theory andh^∗hbe the set of all (unstable) cohomology operations onh^∗.

Definition (I-Kono)

m≥0 (e_m^X)^∗is a split mono ofh^∗h-moduleso

For example, a computation of module weight yields the following.

Theorem (I-Kono)

Mwgt(Spin(9);F₂)≥8whilewgt(Spin(9);F₂)=6.

Leth^∗be a generalised cohomology theory andh^∗hbe the set of all (unstable) cohomology operations onh^∗.

Definition (I-Kono)

m≥0 (e_m^X)^∗is a split mono ofh^∗h-moduleso

For example, a computation of module weight yields the following.

Theorem (I-Kono)

Mwgt(Spin(9);F₂)≥8whilewgt(Spin(9);F₂)=6.

Leth^∗be a generalised cohomology theory andh^∗hbe the set of all (unstable) cohomology operations onh^∗.

Definition (I-Kono)

m≥0 (e_m^X)^∗is a split mono ofh^∗h-moduleso

For example, a computation of module weight yields the following.

Theorem (I-Kono)

Mwgt(Spin(9);F₂)≥8whilewgt(Spin(9);F₂)=6.

Definitions of relative L-S categories

Berstein and Ganea defined their relative L-S categorycat^BG(g)as follows:

Definition (Berstein-Ganea)

For a mapg : K →X,cat^BG(g)is the least numberm≥0such thatKis covered bym+1open subsets,each of which is contractible inX, where we regardgas an inclusiong : K ֒→X.

On the other hand, Fadell and Husseini introduced another version of relative L-S categorycat^FH(K,A)as follows:

Definition (Fadell-Husseini)

For a pair(K,A),cat^FH(K,A)is the least numberm≥0such thatKis covered bym+1open subsetsV ⊃AandU_j,1≤ j≤mwhereVis compressible relativeAintoAand eachUj is contractible inK.

Definitions of relative L-S categories

Berstein and Ganea defined their relative L-S categorycat^BG(g)as follows:

Definition (Berstein-Ganea)

For a mapg : K →X,cat^BG(g)is the least numberm≥0such thatKis covered bym+1open subsets,each of which is contractible inX, where we regardgas an inclusiong : K ֒→X.

On the other hand, Fadell and Husseini introduced another version of relative L-S categorycat^FH(K,A)as follows:

Definition (Fadell-Husseini)

For a pair(K,A),cat^FH(K,A)is the least numberm≥0such thatKis covered bym+1open subsetsV ⊃AandU_j,1≤ j≤mwhereVis compressible relativeAintoAand eachUj is contractible inK.

Definitions of relative L-S categories

Berstein and Ganea defined their relative L-S categorycat^BG(g)as follows:

Definition (Berstein-Ganea)

For a mapg : K →X,cat^BG(g)is the least numberm≥0such thatKis covered bym+1open subsets, each of which is contractible inX, where we regardgas an inclusiong : K ֒→X.

On the other hand, Fadell and Husseini introduced another version of relative L-S categorycat^FH(K,A)as follows:

Definition (Fadell-Husseini)

For a pair(K,A),cat^FH(K,A)is the least numberm≥0such thatKis covered bym+1open subsetsV ⊃AandU_j,1≤ j≤mwhereVis compressible relativeAintoAand eachUj is contractible inK.

Definitions of relative L-S categories

Berstein and Ganea defined their relative L-S categorycat^BG(g)as follows:

Definition (Berstein-Ganea)

For a mapg : K →X,cat^BG(g)is the least numberm≥0such thatKis covered bym+1open subsets, each of which is contractible inX, where we regardgas an inclusiong : K ֒→X.

On the other hand, Fadell and Husseini introduced another version of relative L-S categorycat^FH(K,A)as follows:

Definition (Fadell-Husseini)

For a pair(K,A),cat^FH(K,A)is the least numberm≥0such thatKis covered bym+1open subsetsV ⊃AandU_j,1≤ j≤mwhereVis compressible relativeAintoAand eachUj is contractible inK.

Definitions of relative L-S categories

Berstein and Ganea defined their relative L-S categorycat^BG(g)as follows:

Definition (Berstein-Ganea)

For a mapg : K →X,cat^BG(g)is the least numberm≥0such thatKis covered bym+1open subsets, each of which is contractible inX, where we regardgas an inclusiong : K ֒→X.

On the other hand, Fadell and Husseini introduced another version of relative L-S categorycat^FH(K,A)as follows:

Definition (Fadell-Husseini)

For a pair(K,A),cat^FH(K,A)is the least numberm≥0such thatKis covered bym+1open subsetsV ⊃AandU_j,1≤ j≤mwhereVis compressible relativeAintoAand eachUj is contractible inK.

Definitions of relative L-S categories

Berstein and Ganea defined their relative L-S categorycat^BG(g)as follows:

Definition (Berstein-Ganea)

For a mapg : K →X,cat^BG(g)is the least numberm≥0such thatKis covered bym+1open subsets, each of which is contractible inX, where we regardgas an inclusiong : K ֒→X.

On the other hand, Fadell and Husseini introduced another version of relative L-S categorycat^FH(K,A)as follows:

Definition (Fadell-Husseini)

For a pair(K,A),cat^FH(K,A)is the least numberm≥0such thatKis covered bym+1open subsetsV ⊃AandU_j,1≤ j≤mwhereVis compressible relativeAintoAand eachUj is contractible inK.

More definitions of relative L-S categories

Later, Arkowitz and Lupton introduced yet another version of relative L-S categorycat^AL(h), which can be difined as follows:

Definition (Arkowitz-Lupton)

For a maph : X →Y, letLbe the homotopy fibre ofh. Thencat^AL(h)is the least numberm≥0such thatXis covered bym+1open subsetsV ⊃ L andUj,1≤ j≤mwhereV is compressible intoLand eachUjis contractible inX, where we regardLas the subspace ofX.

Then we may notice that this definition gives a similar but different version of Fadell-Husseini’s relative L-S category for the pair(X,L). Let us denote it bycat^AL(X,L), which is an extended version of Arkowitz-Lupton’s relative L-S categorycat^AL(h).

More definitions of relative L-S categories

Later, Arkowitz and Lupton introduced yet another version of relative L-S categorycat^AL(h), which can be difined as follows:

Definition (Arkowitz-Lupton)

For a maph : X →Y, letLbe the homotopy fibre ofh. Thencat^AL(h)is the least numberm≥0such thatXis covered bym+1open subsetsV ⊃ L andUj,1≤ j≤mwhereV is compressible intoLand eachUjis contractible inX, where we regardLas the subspace ofX.

Then we may notice that this definition gives a similar but different version of Fadell-Husseini’s relative L-S category for the pair(X,L). Let us denote it bycat^AL(X,L), which is an extended version of Arkowitz-Lupton’s relative L-S categorycat^AL(h).

More definitions of relative L-S categories

Later, Arkowitz and Lupton introduced yet another version of relative L-S categorycat^AL(h), which can be difined as follows:

Definition (Arkowitz-Lupton)

For a maph : X →Y, letLbe the homotopy fibre ofh. Thencat^AL(h)is the least numberm≥0such thatXis covered bym+1open subsetsV ⊃ L andUj,1≤ j≤mwhereV is compressible intoLand eachUjis contractible inX, where we regardLas the subspace ofX.

Then we may notice that this definition gives a similar but different version of Fadell-Husseini’s relative L-S category for the pair(X,L). Let us denote it bycat^AL(X,L), which is an extended version of Arkowitz-Lupton’s relative L-S categorycat^AL(h).

More definitions of relative L-S categories

Later, Arkowitz and Lupton introduced yet another version of relative L-S categorycat^AL(h), which can be difined as follows:

Definition (Arkowitz-Lupton)

For a maph : X →Y, letLbe the homotopy fibre ofh. Thencat^AL(h)is the least numberm≥0such thatXis covered bym+1open subsetsV ⊃ L andUj,1≤ j≤mwhereV is compressible intoLand eachUjis contractible inX, where we regardLas the subspace ofX.

Then we may notice that this definition gives a similar but different version of Fadell-Husseini’s relative L-S category for the pair(X,L). Let us denote it bycat^AL(X,L), which is an extended version of Arkowitz-Lupton’s relative L-S categorycat^AL(h).

More definitions of relative L-S categories

Later, Arkowitz and Lupton introduced yet another version of relative L-S categorycat^AL(h), which can be difined as follows:

Definition (Arkowitz-Lupton)

For a maph : X →Y, letLbe the homotopy fibre ofh. Thencat^AL(h)is the least numberm≥0such thatXis covered bym+1open subsetsV ⊃ L andUj,1≤ j≤mwhereV is compressible intoLand eachUjis contractible inX, where we regardLas the subspace ofX.

Then we may notice that this definition gives a similar but different version of Fadell-Husseini’s relative L-S category for the pair(X,L). Let us denote it bycat^AL(X,L), which is an extended version of Arkowitz-Lupton’s relative L-S categorycat^AL(h).

More definitions of relative L-S categories

Later, Arkowitz and Lupton introduced yet another version of relative L-S categorycat^AL(h), which can be difined as follows:

Definition (Arkowitz-Lupton)

For a maph : X →Y, letLbe the homotopy fibre ofh. Thencat^AL(h)is the least numberm≥0such thatXis covered bym+1open subsetsV ⊃ L andUj,1≤ j≤mwhereV is compressible intoLand eachUjis contractible inX, where we regardLas the subspace ofX.

Then we may notice that this definition gives a similar but different version of Fadell-Husseini’s relative L-S category for the pair(X,L). Let us denote it bycat^AL(X,L), which is an extended version of Arkowitz-Lupton’s relative L-S categorycat^AL(h).

Unified version of relative L-S category

To understand these intricate ideas among relative L-S categories and a categorical sequence, we introduce a unified version of a relative L-S category,which explains when the categorical length goes up by one.

Definition

Let(X; K,L:A)be a triad in the category of maps fromA. Then

cat(X; K,L:A)is the least numberm≥0such that the restriction of them+1 fold diagonal map of XtoK,∆^m+1|_K : K →Q_m+1

X, is compressible relativeAintoT^m+1(X,L)=L×Q_m

X∪X×T^mX the relative fat wedge.

Theorem

1 cat^BG(X,K)=cat(X; K,∗:∗), cat^FH(X,A)=cat(X; X,A:A),

2 cat^AL(X,L)=cat(X; X,L:∗).

Unified version of relative L-S category

To understand these intricate ideas among relative L-S categories and a categorical sequence, we introduce a unified version of a relative L-S category, which explains when the categorical length goes up by one.

Definition

Let(X; K,L:A)be a triad in the category of maps fromA. Then

cat(X; K,L:A)is the least numberm≥0such that the restriction of them+1 fold diagonal map of XtoK,∆^m+1|_K : K →Q_m+1

X,is compressible relativeAintoT^m+1(X,L)=L×Q_m

X∪X×T^mX the relative fat wedge.

Theorem

1 cat^BG(X,K)=cat(X; K,∗:∗), cat^FH(X,A)=cat(X; X,A:A),

2 cat^AL(X,L)=cat(X; X,L:∗).

Unified version of relative L-S category

To understand these intricate ideas among relative L-S categories and a categorical sequence, we introduce a unified version of a relative L-S category, which explains when the categorical length goes up by one.

Definition

Let(X; K,L:A)be a triad in the category of maps fromA. Then

cat(X; K,L:A)is the least numberm≥0such that the restriction of them+1 fold diagonal map of XtoK,∆^m+1|_K : K →Q_m+1

X,is compressible relativeAintoT^m+1(X,L)=L×Q_m

X∪X×T^mX the relative fat wedge.

Theorem

1 cat^BG(X,K)=cat(X; K,∗:∗), cat^FH(X,A)=cat(X; X,A:A),

2 cat^AL(X,L)=cat(X; X,L:∗).

Unified version of relative L-S category

To understand these intricate ideas among relative L-S categories and a categorical sequence, we introduce a unified version of a relative L-S category, which explains when the categorical length goes up by one.

Definition

Let(X; K,L:A)be a triad in the category of maps fromA. Then

cat(X; K,L:A)is the least numberm≥0such that the restriction of them+1 fold diagonal map of XtoK,∆^m+1|_K : K →Q_m+1

X,is compressible relativeAintoT^m+1(X,L)=L×Q_m

X∪X×T^mX the relative fat wedge.

Theorem

1 cat^BG(X,K)=cat(X; K,∗:∗), cat^FH(X,A)=cat(X; X,A:A),

2 cat^AL(X,L)=cat(X; X,L:∗).

Unified version of relative L-S category

To understand these intricate ideas among relative L-S cat