characteristic classes and invariants of spin geometryxbli/haibao duan.pdf · characteristic...

Characteristic classes and Invariants of Spin Geometry

Haibao Duan

Institue of Mathematics, CAS

2018 Workshop on Algebraic and Geometric Topology,Southwest Jiaotong University

July 29, 2018

Haibao Duan (CAS) Characteristic classes and Invariants of Spin Geometry July 29, 2018 1 / 25

The arrangement of the talk

1 The problem and its background

2 Main results

3 Applications

4 The proof of main result

The problem and its background

The spin group Spin(n) is the universal covering of the special orthogonalgroup SO(n).The spinc(n) group Spinc(n) is the central extension Spin(n)×Z2 U(1) ofSO(n) by the circle group U(1).

In this talk I will

introduce a pair F = {γ, α} of cohomology operations;

construct the integral cohomology rings of the classifying spacesBSpinc (n) and BSpin(n) by using these operations.

In this talk I will

Motivation: Assume that a minimal set {q1, · · · , qr} of generators ofthe ring H∗(BSpin(n)) has been specified.

We can

1 define the spin characteristic classes for a spin bundle ξ over aspace X with classifying map f : X → BSpin(n) by setting

qi (ξ) = f ∗(qi ) ∈ H∗(X ), 1 ≤ i ≤ r ;

2 obtain the basic Weyl invariants of the group Spin(n) by putting

ck = i∗(qk) ∈ H∗(BT )W = Z [t1, · · · , tm]W , 1 ≤ k ≤ r

where i : BT → BSpin(n) is induced by the inclusion of a maximaltorus T ⊂ Spin(n), and where m = dimT .

Motivation: Assume that a minimal set {q1, · · · , qr} of generators ofthe ring H∗(BSpin(n)) has been specified. We can

qi (ξ) = f ∗(qi ) ∈ H∗(X ), 1 ≤ i ≤ r ;

Motivation: Assume that a minimal set {q1, · · · , qr} of generators ofthe ring H∗(BSpin(n)) has been specified. We can

qi (ξ) = f ∗(qi ) ∈ H∗(X ), 1 ≤ i ≤ r ;

Since the discovery of spinors by Cartan in 1913, the spin structure onRiemannian manifolds has found significant and wide applications togeometry and mathematical physics;

However, a precise definition of spin structure was possible only after thenotion of fiber bundle had been introduced

Haefliger (1956) found that the second Stiefel Whitney class w2(M)is the only obstruction to the existence of a spin structure on anorientable Riemannian manifold M.

This was extended by Borel and Hirzebruch (1958) to cases of vectorbundles, and by Karoubi (1968) to the non-orientablepseudo-Riemannian manifolds.

The problem and background

Earlier works on the problem:

The mod 2 cohomology of the space BSpin(n) was computed by Borel(1953) for n ≤ 10, and was completed by Quillen (1972) for all n.

Thomas (1962) calculated the integral cohomology of BSpin(∞) in thestable range, but the result was subject to the choice of two sets{Φi ,Ψi} of indeterminats.

In the context of Weyl invariants, a description of the integralcohomology H∗(BSpin(n)) was formulated by Benson and Wood(1995), where explicit generators and relations are absent:

”We have not set about the rather daunting task of using thisdescription to give explicit generators and relations”

由扫描全能王扫描创建

In mathematical physics the Postnikov tower anchored by the space BSO(n)

· · · → BFivebrane(n) → BString(n) → BSpin(n) → BSO(n).

It is expected that the operations F = {γ, α} introduced in the talk willalso be useful to construct the integral cohomology rings of the furtherspaces BString(n),BFivebrane(n), · · · in the tower.

In mathematical physics the Postnikov tower anchored by the space BSO(n)

· · · → BFivebrane(n) → BString(n) → BSpin(n) → BSO(n).

It is expected that the operations F = {γ, α} introduced in the talk willalso be useful to construct the integral cohomology rings of the furtherspaces BString(n),BFivebrane(n), · · · in the tower.

Main results

For any topological space X and a cohomology class u ∈ H2r (X ;Z2) thereholds the following universal relations:

δ2(u ∪ u) = 2δ4(B(u)) ∈ H4r+1(X )

where B : H2r (X ;Z2)→ H4r (X ;Z4) denotes the Pontryagin square.

Definition

The space X is called δ2 formal if δ2(u ∪ u) = 0 for all u ∈ H2r (X ;Z2).

Corollary

If X is a space whose integral cohomologies H4r+1(X ), r ≥ 1, has notorsion element of order 4, then X is δ2 formal.

In particular, all the 1 connected Lie groups, the classifying spaces BSO(n)

and BSpin(n) , as well as the Thom spectrum MO(n), n ≥ 1, are examplesof the δ2 formal spaces.

Main results

δ2(u ∪ u) = 2δ4(B(u)) ∈ H4r+1(X )

Definition

Corollary

Main results

δ2(u ∪ u) = 2δ4(B(u)) ∈ H4r+1(X )

Definition

Corollary

Main results

δ2(u ∪ u) = 2δ4(B(u)) ∈ H4r+1(X )

Definition

Corollary

Main results

Recall the Bockstein operator Sq1 = ρ2 ◦ δ2 on the algebra H∗(X ;Z2)defines the decomposition

H∗(X ;Z2) = ker Sq1 ⊕ S∗2 (X ) with S∗2 (X ) = H∗(X ;Z2)/ ker Sq1.

Theorem 1

Let X be a δ2 formal space. There exists a unique pair of cohomologicaloperations

F : H2r (X ;Z2)→ S4r2 (X ;Z2)× H4r (X ;Z4),

written F (u) = (γ(u), α(u)), that satisfies the following properties

i) α(u) ∈ Im ρ4;ii) B(u) = α(u) + θ(γ(u));iii) Sq1(γ(u)) = Sq2rSq1(u) + u ∪ Sq1(u).

Main results

Theorem 1

F : H2r (X ;Z2)→ S4r2 (X ;Z2)× H4r (X ;Z4),

Main results

Theorem 1

F : H2r (X ;Z2)→ S4r2 (X ;Z2)× H4r (X ;Z4),

Main results

Since γ(u) ∈ S∗2 (X ) while Sq1 injects on S∗2 (X ), the operation γ ischaracterized uniquely by the equation iii):

Sq1(γ(u)) = Sq2rSq1(u) + u ∪ Sq1(u).

This operation γ can be iterrated to yield the following notion.

Definition

Given an even degree cohomology class u ∈ H2r (X ;Z2) of a δ2-formalspace X , the sequence

{u, u(1), u(2), · · ·

}of elements with

u(1) = γ(u), u(k+1) = γ(u(k)),

is called the derived sequence of the initial class u.

Main results

Since γ(u) ∈ S∗2 (X ) while Sq1 injects on S∗2 (X ), the operation γ ischaracterized uniquely by the equation iii):

Sq1(γ(u)) = Sq2rSq1(u) + u ∪ Sq1(u).

This operation γ can be iterrated to yield the following notion.

Definition

Given an even degree cohomology class u ∈ H2r (X ;Z2) of a δ2-formalspace X , the sequence

{u, u(1), u(2), · · ·

}of elements with

u(1) = γ(u), u(k+1) = γ(u(k)),

is called the derived sequence of the initial class u.

Main results

Example

For the δ2 formal space X = BSO(n) we take

u = w2 ∈ H∗(BSO(n);Z2) = Z2[w2, · · · ,wn].

By the ”coefficients comparison” method we get

γ(w2) = w(1)2 = w4,

γ2(w2) = w(2)2 = w8 + w2w6,

...γk(w2) = w

(k)2 = w2k + w2w2k−2 + · · ·+ w2k−1−2w2k−1+2

+ terms with order ≥ 3.

These imply, in contrast to the solution to the Peterson’s hit problem forH∗(BSO(n),Z2) over the Steenrod algebra, that

{w2,w(1)2 ,w

(2)2 , · · · } ≡ {w2,w4,w8, · · · } mod decompositables.

Main results

Example

u = w2 ∈ H∗(BSO(n);Z2) = Z2[w2, · · · ,wn].

γ(w2) = w(1)2 = w4,

γ2(w2) = w(2)2 = w8 + w2w6,

...γk(w2) = w

(k)2 = w2k + w2w2k−2 + · · ·+ w2k−1−2w2k−1+2

{w2,w(1)2 ,w

Main results

Example

u = w2 ∈ H∗(BSO(n);Z2) = Z2[w2, · · · ,wn].

γ(w2) = w(1)2 = w4,

γ2(w2) = w(2)2 = w8 + w2w6,

...γk(w2) = w

(k)2 = w2k + w2w2k−2 + · · ·+ w2k−1−2w2k−1+2

{w2,w(1)2 ,w

Main results

Returning to the operator α in Theorem 1, the relation i) implies that italways admits an integral lift f

α(u) ∈ Imρ4 ⇔H4r (X )

f ↗ ↓ ρ4H2r (X ;Z2)

α→ H4r (X ;Z4)

In the case X = BSO(n) a canonical choice of an integral lift f can beeasily formulated. Recall from Feshbach and Brown (1983) that

1 H∗(BSO(n)) =

{Z[p1, p2, · · · , p[ n−1

2 ], en]⊕ τ2(BSO(n)) if n is even;

Z[p1, p2, · · · , p[ n−12 ]]⊕ τ2(BSO(n)) if n is odd,

In view of this presentation we can define an integral lift of α

f : H∗(BSO(n);Z2)→ H∗(BSO(n))

Main results

α(u) ∈ Imρ4 ⇔H4r (X )

f ↗ ↓ ρ4H2r (X ;Z2)

α→ H4r (X ;Z4)

In the case X = BSO(n) a canonical choice of an integral lift f can beeasily formulated.

Recall from Feshbach and Brown (1983) that

1 H∗(BSO(n)) =

{Z[p1, p2, · · · , p[ n−1

Main results

α(u) ∈ Imρ4 ⇔H4r (X )

f ↗ ↓ ρ4H2r (X ;Z2)

α→ H4r (X ;Z4)

1 H∗(BSO(n)) =

{Z[p1, p2, · · · , p[ n−1

Main results

α(u) ∈ Imρ4 ⇔H4r (X )

f ↗ ↓ ρ4H2r (X ;Z2)

α→ H4r (X ;Z4)

1 H∗(BSO(n)) =

{Z[p1, p2, · · · , p[ n−1

Main results

by the following practical rules:

1 in accordance to u = w2r ,w2r+1 or u = wn when n is even, definef (u) := pr , δ2(Sq2rw2r+1) or e2n ;

2 f (u) := f (wi1) · · · f (wik ) if u = wi1 · · ·wik ;

3 f (u) := f (u1) + · · ·+ f (uk) if u = u1 + · · ·+ uk with ui ’s distinctmonomials in w2, · · · ,wn.

Based on Theorem 1 it can be shown that

Theorem 2

The pair (f , γ) of operations satisfies the following properties: for anyu ∈ H2r (BSO(n);Z2) one has

i) B(u) = ρ4(f (u)) + θ(γ(u));ii) Sq1(γ(u)) = Sq2rSq1(u) + u ∪ Sq1(u).

Main results

Theorem 2

Main results

Theorem 2

Main results

Theorem 2

Main results

Example

Take u = w2r . Then f (w2r ) = pr by the definition of f . Solving theequation ii) by the coefficients comparison method gives

γ(w2r ) = w4r + w2w4r−2 + · · ·+ w2r−2w2r+2.

Substituting these into the formula i) of Theorem 2 yields that

B(w2r ) = ρ4(pr ) + θ(w4r + w2w4r−2 + · · ·+ w2r−2w2r+2).

This formula was first obtained by W.T.Wu (On th Pontryagin classesI,II, III, Acta. Sinica, 1953-54) by computing with the cochaincomplex associated to the Schubert cells decomposition on BSO(n).

S.S. Chern suggested a different approach to the formula, which wasimplemented by Thomas (Trans. AMS, 1960).

Main results

Example

γ(w2r ) = w4r + w2w4r−2 + · · ·+ w2r−2w2r+2.Substituting these into the formula i) of Theorem 2 yields that

Main results

Example

Main results

Example

Main results

For each n ≥ 8 we set h(n) =[n−12

]and let {w2,w

(1)2 , · · · ,w (h(n)−1)

2 } bethe first h(n) terms of the derived sequence of w2.

Applying the operatorf of Theorem 2 we get the sequence of integral cohomology classes:

{f (w2), f (w(1)2 ), · · · , f (w

(h(n)−1)2 )} ∈ H∗(BSO(n)).

In view of the fibration CP∞i→ BSpinc (n)

π→ BSO(n) we can show

Theorem 3

There is a unique set {q, qr , 1 ≤ r ≤ h(n)− 1} of integral cohomologyclasses on BSpinc (n), degqr = 2r+1, that satisfies the following system:

1 ρ2(q) = π∗w2, ρ2(qr ) = π∗w(r)2 ;

2 2q1 − q2 = π∗p1, 2qr+1 − q2r = π∗f (w

(r)2 ).

Main results

]and let {w2,w

(1)2 , · · · ,w (h(n)−1)

2 } bethe first h(n) terms of the derived sequence of w2. Applying the operatorf of Theorem 2 we get the sequence of integral cohomology classes:

{f (w2), f (w(1)2 ), · · · , f (w

(h(n)−1)2 )} ∈ H∗(BSO(n)).

Theorem 3

1 ρ2(q) = π∗w2, ρ2(qr ) = π∗w(r)2 ;

2 2q1 − q2 = π∗p1, 2qr+1 − q2r = π∗f (w

(r)2 ).

Main results

]and let {w2,w

(1)2 , · · · ,w (h(n)−1)

{f (w2), f (w(1)2 ), · · · , f (w

(h(n)−1)2 )} ∈ H∗(BSO(n)).

Theorem 3

1 ρ2(q) = π∗w2, ρ2(qr ) = π∗w(r)2 ;

2 2q1 − q2 = π∗p1, 2qr+1 − q2r = π∗f (w

(r)2 ).

Main results

]and let {w2,w

(1)2 , · · · ,w (h(n)−1)

{f (w2), f (w(1)2 ), · · · , f (w

(h(n)−1)2 )} ∈ H∗(BSO(n)).

Theorem 3

1 ρ2(q) = π∗w2, ρ2(qr ) = π∗w(r)2 ;

2 2q1 − q2 = π∗p1, 2qr+1 − q2r = π∗f (w

(r)2 ).

Main results

Regarding H∗(BSpinc (n)) as a module over its subring π∗H∗(BSO(n)) wehave

Theorem 4

The cohomology ring H∗(BSpinc (n)) has the presentation

H∗(BSpinc (n)) = π∗H∗(BSO(n))⊗ Z[δ]⊗∆(q, q1, · · · , qh(n)−1)

that is subject to the following relations

2q1 − q2 = π∗p1, 2qr+1 − q2r = π∗f (w

(r)2 ), 4δ − q2

h(n)−1 = h.

∆(q, q1, · · · , qh(n)−1) denotes the free Z module in the simple systemq, q1, · · · , qh(n)−1 of generators (in Borel’s notation);

δ is the Euler class of the complex spin representationSpinc(n)→ U(2h(n)) (Atiyah and Bott, 1964).

Main results

Regarding H∗(BSpinc (n)) as a module over its subring π∗H∗(BSO(n)) wehave

Theorem 4

The cohomology ring H∗(BSpinc (n)) has the presentation

H∗(BSpinc (n)) = π∗H∗(BSO(n))⊗ Z[δ]⊗∆(q, q1, · · · , qh(n)−1)

that is subject to the following relations

2q1 − q2 = π∗p1, 2qr+1 − q2r = π∗f (w

(r)2 ), 4δ − q2

h(n)−1 = h.

∆(q, q1, · · · , qh(n)−1) denotes the free Z module in the simple systemq, q1, · · · , qh(n)−1 of generators (in Borel’s notation);

δ is the Euler class of the complex spin representationSpinc(n)→ U(2h(n)) (Atiyah and Bott, 1964).

Main results

Computing with the Gysin sequence of the circle fibration

U(1) ↪→ BSpin(n) → BSpinc (n) → CP∞

shows the following result, where π : BSpin(n) → BSO(n) is induced by thecovering Spin(n)→ SO(n).

Theorem 5

The cohomology H∗(BSpin(n)) has the presentation

H∗(BSpin(n)) = π∗H∗(BSO(n))⊗∆(q1, · · · , qh(n)−1, δ±),

subject to the following relations:

2q1 = p1, 2qr+1 − q2r = π∗f (w

(r)2 ), · · · .

where δ± denotes the Euler class of the real spin (or the half spin ±)representation of the group Spin(n) depending on the values of n mod8

Main results

Computing with the Gysin sequence of the circle fibration

U(1) ↪→ BSpin(n) → BSpinc (n) → CP∞

shows the following result, where π : BSpin(n) → BSO(n) is induced by thecovering Spin(n)→ SO(n).

Theorem 5

The cohomology H∗(BSpin(n)) has the presentation

H∗(BSpin(n)) = π∗H∗(BSO(n))⊗∆(q1, · · · , qh(n)−1, δ±),

subject to the following relations:

2q1 = p1, 2qr+1 − q2r = π∗f (w

(r)2 ), · · · .

where δ± denotes the Euler class of the real spin (or the half spin ±)representation of the group Spin(n) depending on the values of n mod8

Applications

1 The spin characteristic classes are more subtle that the Pontyaginclasses: there exist spin vector bundles ξ for which

pi (ξ) = 0, wi (ξ) = 0, i ≥ 1,

but qr (ξ) 6= 0 for some r ≥ 1 (Thomas, 1962);2 In the Milnor’s calculation (1957) on the group Θ7 if one uses the

Spin characteristic classes q1, q2 in place of the Pontryagin classesp1, p2, one obtains |Θ7| ≥ 14 instead of |Θ7| ≥ 7;

3 For an 8 dimensional manifold M, a pair of integral cohomologyclasses (a, b) can be realized as the first two spin characteristic classesof a stable spin bundle on M, if and only if

a2 + b ≡ U13 ∪ a on H8(M; Z3)

where U13 is the mod3 Wu class of M (Duan, 1991).

For such realization problem in the unstable instances, there shouldbe more relations in certain characteristic classes.

Applications

pi (ξ) = 0, wi (ξ) = 0, i ≥ 1,

but qr (ξ) 6= 0 for some r ≥ 1 (Thomas, 1962);

2 In the Milnor’s calculation (1957) on the group Θ7 if one uses theSpin characteristic classes q1, q2 in place of the Pontryagin classesp1, p2, one obtains |Θ7| ≥ 14 instead of |Θ7| ≥ 7;

a2 + b ≡ U13 ∪ a on H8(M; Z3)

Applications

pi (ξ) = 0, wi (ξ) = 0, i ≥ 1,

a2 + b ≡ U13 ∪ a on H8(M; Z3)

Applications

pi (ξ) = 0, wi (ξ) = 0, i ≥ 1,

a2 + b ≡ U13 ∪ a on H8(M; Z3)

Applications

pi (ξ) = 0, wi (ξ) = 0, i ≥ 1,

a2 + b ≡ U13 ∪ a on H8(M; Z3)

For such realization problem in the unstable instances, there shouldbe more relations in certain characteristic classes.Haibao Duan (CAS) Characteristic classes and Invariants of Spin Geometry July 29, 2018 20 / 25

Applications

1 There is a simple recurrence to produced the basic W = WSpin(n)

invariants

ck = i∗(qk) ∈ H∗(BT )W = Z [t1, · · · , tm]W , 1 ≤ k ≤ h(n).

2 The relations on the cohomology H∗(BSpin(n))

2qr+1 − q2r = π∗f (w

(r)2 )

indicates that the ring H∗(BT )W of integral W = WSpin(n) invariantsis not a polynomial ring.

Applications

1 There is a simple recurrence to produced the basic W = WSpin(n)

invariants

ck = i∗(qk) ∈ H∗(BT )W = Z [t1, · · · , tm]W , 1 ≤ k ≤ h(n).

2 The relations on the cohomology H∗(BSpin(n))

2qr+1 − q2r = π∗f (w

(r)2 )

indicates that the ring H∗(BT )W of integral W = WSpin(n) invariantsis not a polynomial ring.

The proof of main results

We conclude with a proof of the main result.

Theorem 1. Let X be a δ2 formal space. There exists a unique pair ofcohomological operations

F : H2r (X ;Z2)→ S4r2 (X ;Z2)× H4r (X ;Z4),

written F (u) = (γ(u), α(u)), that satisfies the following properties:

i) α(u) ∈ Im ρ4;

ii) B(u) = α(u) + θ(γ(u));

iii) Sq1(γ(u)) = Sq2rSq1(u) + u ∪ Sq1(u).

For any topological space X and a cohomology class u ∈ H2r (X ;Z2) therehold the following universal relations:

(4.1) δ2(u ∪ u) = 2δ4(B(u)) in H4n+1(X );(4.2) ρ2δ4B(u) = Sq2nSq1u + u ∪ Sq1u in H4n+1(X ;Z2).

Assume that the space X is δ2 formal. Then δ4(B(u)) ∈ Im δ2 by (4.1). Inview of the isomorphism

δ2 : S4r2 (X ) ∼= Im δ2

there exists a unique element u1 ∈ S4r2 (X ) so that

(4.3) δ2(u1) = δ4(B(u)).

We can now formulate the desired operations

F = (γ, α) : H2r (X ;Z2)→ S4r2 (X )⊗ H4r (X ;Z4)

by setting

(4.4) γ(u) := u1, α(u) := B(u)− θ(γ(u)).

Assume that the space X is δ2 formal. Then δ4(B(u)) ∈ Im δ2 by (4.1).

Inview of the isomorphism

δ2 : S4r2 (X ) ∼= Im δ2

(4.3) δ2(u1) = δ4(B(u)).

F = (γ, α) : H2r (X ;Z2)→ S4r2 (X )⊗ H4r (X ;Z4)

by setting

(4.4) γ(u) := u1, α(u) := B(u)− θ(γ(u)).

δ2 : S4r2 (X ) ∼= Im δ2

(4.3) δ2(u1) = δ4(B(u)).

F = (γ, α) : H2r (X ;Z2)→ S4r2 (X )⊗ H4r (X ;Z4)

by setting

(4.4) γ(u) := u1, α(u) := B(u)− θ(γ(u)).

δ2 : S4r2 (X ) ∼= Im δ2

(4.3) δ2(u1) = δ4(B(u)).

F = (γ, α) : H2r (X ;Z2)→ S4r2 (X )⊗ H4r (X ;Z4)

by setting

(4.4) γ(u) := u1, α(u) := B(u)− θ(γ(u)).

Applying ρ2 to both sides of (4.3) we get by (4.2) that

Sq1γ(u) = Sq2nSq1u + u ∪ Sq1u.

Moreover, from

δ4α(u) = δ4(B(u)− θ(γ(u)))= δ4(B(u))− δ2(γ(u)) (by δ4 ◦ θ = δ2)= 0 (by (4.3))

we find that α(u) ∈ Im ρ4.

Summarizing, we have obtained the operation F that satisfies theproperties i), ii) and iii) of Theorem 1, whose uniqueness comes from itsdefinition (4.4).�

Moreover, from

Thanks!

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