Characteristic classes and Invariants of Spin Geometry

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 Haibao Duan (CAS) Characteristic classes and Invariants of Spin Geometry July 29, 2018 2 / 25

The problem and its background The spin group Spin(n) is the universal covering of the special orthogonal group SO(n). The spin c (n) group Spin c (n) is the central extension Spin(n) Z2 U(1) of SO(n) by the circle group U(1). In this talk I will introduce a pair F = {γ, α} of cohomology operations; Haibao Duan (CAS) Characteristic classes and Invariants of Spin Geometry July 29, 2018 3 / 25

The problem and its background The spin group Spin(n) is the universal covering of the special orthogonal group SO(n). The spin c (n) group Spin c (n) is the central extension Spin(n) Z2 U(1) of SO(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 spaces B Spin c (n) and B Spin(n) by using these operations. Haibao Duan (CAS) Characteristic classes and Invariants of Spin Geometry July 29, 2018 3 / 25

The problem and its background Motivation: Assume that a minimal set {q 1,, q r } of generators of the ring H (B Spin(n) ) has been specified. Haibao Duan (CAS) Characteristic classes and Invariants of Spin Geometry July 29, 2018 4 / 25

The problem and its background Motivation: Assume that a minimal set {q 1,, q r } of generators of the ring H (B Spin(n) ) has been specified. We can 1 define the spin characteristic classes for a spin bundle ξ over a space X with classifying map f : X B Spin(n) by setting q i (ξ) = f (q i ) H (X ), 1 i r; 2 obtain the basic Weyl invariants of the group Spin(n) by putting c k = i (q k ) H (B T ) W = Z[t 1,, t m ] W, 1 k r where i : B T B Spin(n) is induced by the inclusion of a maximal torus T Spin(n), and where m = dimt. Haibao Duan (CAS) Characteristic classes and Invariants of Spin Geometry July 29, 2018 4 / 25

The problem and its background Since the discovery of spinors by Cartan in 1913, the spin structure on Riemannian manifolds has found significant and wide applications to geometry and mathematical physics; However, a precise definition of spin structure was possible only after the notion of fiber bundle had been introduced Haefliger (1956) found that the second Stiefel Whitney class w 2 (M) is the only obstruction to the existence of a spin structure on an orientable Riemannian manifold M. Haibao Duan (CAS) Characteristic classes and Invariants of Spin Geometry July 29, 2018 5 / 25

The problem and its background Since the discovery of spinors by Cartan in 1913, the spin structure on Riemannian manifolds has found significant and wide applications to geometry and mathematical physics; However, a precise definition of spin structure was possible only after the notion of fiber bundle had been introduced Haefliger (1956) found that the second Stiefel Whitney class w 2 (M) is the only obstruction to the existence of a spin structure on an orientable Riemannian manifold M. This was extended by Borel and Hirzebruch (1958) to cases of vector bundles, and by Karoubi (1968) to the non-orientable pseudo-riemannian manifolds. Haibao Duan (CAS) Characteristic classes and Invariants of Spin Geometry July 29, 2018 5 / 25

The problem and background Earlier works on the problem: The mod 2 cohomology of the space B Spin(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 B Spin( ) in the stable range, but the result was subject to the choice of two sets {Φ i, Ψ i } of indeterminats. In the context of Weyl invariants, a description of the integral cohomology H (B Spin(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 this description to give explicit generators and relations Haibao Duan (CAS) Characteristic classes and Invariants of Spin Geometry July 29, 2018 6 / 25

The problem and its background Haibao Duan (CAS) Characteristic classes and Invariants of Spin Geometry July 29, 2018 7 / 25

The problem and its background Haibao Duan (CAS) Characteristic classes and Invariants of Spin Geometry July 29, 2018 8 / 25

The problem and its background In mathematical physics the Postnikov tower anchored by the space B SO(n) reads B Fivebrane(n) B String(n) B Spin(n) B SO(n). Haibao Duan (CAS) Characteristic classes and Invariants of Spin Geometry July 29, 2018 9 / 25

The problem and its background In mathematical physics the Postnikov tower anchored by the space B SO(n) reads B Fivebrane(n) B String(n) B Spin(n) B SO(n). It is expected that the operations F = {γ, α} introduced in the talk will also be useful to construct the integral cohomology rings of the further spaces B String(n), B Fivebrane(n), in the tower. Haibao Duan (CAS) Characteristic classes and Invariants of Spin Geometry July 29, 2018 9 / 25

Main results For any topological space X and a cohomology class u H 2r (X ; Z 2 ) there holds the following universal relations: δ 2 (u u) = 2δ 4 (B(u)) H 4r+1 (X ) where B : H 2r (X ; Z 2 ) H 4r (X ; Z 4 ) denotes the Pontryagin square. Definition The space X is called δ 2 formal if δ 2 (u u) = 0 for all u H 2r (X ; Z 2 ). Corollary If X is a space whose integral cohomologies H 4r+1 (X ), r 1, has no torsion element of order 4, then X is δ 2 formal. Haibao Duan (CAS) Characteristic classes and Invariants of Spin Geometry July 29, 2018 10 / 25

Main results For any topological space X and a cohomology class u H 2r (X ; Z 2 ) there holds the following universal relations: δ 2 (u u) = 2δ 4 (B(u)) H 4r+1 (X ) where B : H 2r (X ; Z 2 ) H 4r (X ; Z 4 ) denotes the Pontryagin square. Definition The space X is called δ 2 formal if δ 2 (u u) = 0 for all u H 2r (X ; Z 2 ). Corollary If X is a space whose integral cohomologies H 4r+1 (X ), r 1, has no torsion element of order 4, then X is δ 2 formal. In particular, all the 1 connected Lie groups, the classifying spaces B SO(n) and B Spin(n), as well as the Thom spectrum MO(n), n 1, are examples of the δ 2 formal spaces. Haibao Duan (CAS) Characteristic classes and Invariants of Spin Geometry July 29, 2018 10 / 25

Main results Recall the Bockstein operator Sq 1 = ρ 2 δ 2 on the algebra H (X ; Z 2 ) defines the decomposition H (X ; Z 2 ) = ker Sq 1 S 2 (X ) with S 2 (X ) = H (X ; Z 2 )/ ker Sq 1. Haibao Duan (CAS) Characteristic classes and Invariants of Spin Geometry July 29, 2018 11 / 25

Main results Recall the Bockstein operator Sq 1 = ρ 2 δ 2 on the algebra H (X ; Z 2 ) defines the decomposition H (X ; Z 2 ) = ker Sq 1 S 2 (X ) with S 2 (X ) = H (X ; Z 2 )/ ker Sq 1. Theorem 1 Let X be a δ 2 operations formal space. There exists a unique pair of cohomological F : H 2r (X ; Z 2 ) S 4r 2 (X ; Z 2) H 4r (X ; Z 4 ), written F (u) = (γ(u), α(u)), that satisfies the following properties Haibao Duan (CAS) Characteristic classes and Invariants of Spin Geometry July 29, 2018 11 / 25

Main results Recall the Bockstein operator Sq 1 = ρ 2 δ 2 on the algebra H (X ; Z 2 ) defines the decomposition H (X ; Z 2 ) = ker Sq 1 S 2 (X ) with S 2 (X ) = H (X ; Z 2 )/ ker Sq 1. Theorem 1 Let X be a δ 2 operations formal space. There exists a unique pair of cohomological F : H 2r (X ; Z 2 ) S 4r 2 (X ; Z 2) H 4r (X ; Z 4 ), written F (u) = (γ(u), α(u)), that satisfies the following properties i) α(u) Im ρ 4 ; ii) B(u) = α(u) + θ(γ(u)); iii) Sq 1 (γ(u)) = Sq 2r Sq 1 (u) + u Sq 1 (u). Haibao Duan (CAS) Characteristic classes and Invariants of Spin Geometry July 29, 2018 11 / 25

Main results Since γ(u) S2 (X ) while Sq1 injects on S2 (X ), the operation γ is characterized uniquely by the equation iii): Sq 1 (γ(u)) = Sq 2r Sq 1 (u) + u Sq 1 (u). Haibao Duan (CAS) Characteristic classes and Invariants of Spin Geometry July 29, 2018 12 / 25

Main results Since γ(u) S2 (X ) while Sq1 injects on S2 (X ), the operation γ is characterized uniquely by the equation iii): Sq 1 (γ(u)) = Sq 2r Sq 1 (u) + u Sq 1 (u). This operation γ can be iterrated to yield the following notion. Definition Given an even degree cohomology class u H 2r (X ; Z 2 ) of a δ 2 -formal space 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. Haibao Duan (CAS) Characteristic classes and Invariants of Spin Geometry July 29, 2018 12 / 25

Main results Example For the δ 2 formal space X = B SO(n) we take u = w 2 H (B SO(n) ; Z 2 ) = Z 2 [w 2,, w n ]. Haibao Duan (CAS) Characteristic classes and Invariants of Spin Geometry July 29, 2018 13 / 25

Main results Example For the δ 2 formal space X = B SO(n) we take u = w 2 H (B SO(n) ; Z 2 ) = Z 2 [w 2,, w n ]. By the coefficients comparison method we get γ(w 2 ) = w (1) 2 = w 4, γ 2 (w 2 ) = w (2) 2 = w 8 + w 2 w 6,. γ k (w 2 ) = w (k) 2 = w 2 k + w 2 w 2 k 2 + + w 2 k 1 2w 2 k 1 +2 + terms with order 3. Haibao Duan (CAS) Characteristic classes and Invariants of Spin Geometry July 29, 2018 13 / 25

Main results Example For the δ 2 formal space X = B SO(n) we take u = w 2 H (B SO(n) ; Z 2 ) = Z 2 [w 2,, w n ]. By the coefficients comparison method we get γ(w 2 ) = w (1) 2 = w 4, γ 2 (w 2 ) = w (2) 2 = w 8 + w 2 w 6,. γ k (w 2 ) = w (k) 2 = w 2 k + w 2 w 2 k 2 + + w 2 k 1 2w 2 k 1 +2 + terms with order 3. These imply, in contrast to the solution to the Peterson s hit problem for H (B SO(n), Z 2 ) over the Steenrod algebra, that {w 2, w (1) 2, w (2) 2, } {w 2, w 4, w 8, } mod decompositables. Haibao Duan (CAS) Characteristic classes and Invariants of Spin Geometry July 29, 2018 13 / 25

Main results Returning to the operator α in Theorem 1, the relation i) implies that it always admits an integral lift f α(u) Imρ 4 H 2r (X ; Z 2 ) H 4r (X ) f ρ 4 α H 4r (X ; Z 4 ) Haibao Duan (CAS) Characteristic classes and Invariants of Spin Geometry July 29, 2018 14 / 25

Main results Returning to the operator α in Theorem 1, the relation i) implies that it always admits an integral lift f α(u) Imρ 4 H 2r (X ; Z 2 ) H 4r (X ) f ρ 4 α H 4r (X ; Z 4 ) In the case X = B SO(n) a canonical choice of an integral lift f can be easily formulated. Haibao Duan (CAS) Characteristic classes and Invariants of Spin Geometry July 29, 2018 14 / 25

Main results Returning to the operator α in Theorem 1, the relation i) implies that it always admits an integral lift f α(u) Imρ 4 H 2r (X ; Z 2 ) H 4r (X ) f ρ 4 α H 4r (X ; Z 4 ) In the case X = B SO(n) a canonical choice of an integral lift f can be easily formulated. Recall from Feshbach and Brown (1983) that { Z[p1, p 2,, p 1 H [ n 1 2 ] (B SO(n) ) =, e n] τ 2 (B SO(n) ) if n is even; Z[p 1, p 2,, p [ n 1 2 ] ] τ 2(B SO(n) ) if n is odd, Haibao Duan (CAS) Characteristic classes and Invariants of Spin Geometry July 29, 2018 14 / 25

Main results Returning to the operator α in Theorem 1, the relation i) implies that it always admits an integral lift f α(u) Imρ 4 H 2r (X ; Z 2 ) H 4r (X ) f ρ 4 α H 4r (X ; Z 4 ) In the case X = B SO(n) a canonical choice of an integral lift f can be easily formulated. Recall from Feshbach and Brown (1983) that { Z[p1, p 2,, p 1 H [ n 1 2 ] (B SO(n) ) =, e n] τ 2 (B SO(n) ) if n is even; Z[p 1, p 2,, p [ n 1 2 ] ] τ 2(B SO(n) ) if n is odd, In view of this presentation we can define an integral lift of α f : H (B SO(n) ; Z 2 ) H (B SO(n) ) Haibao Duan (CAS) Characteristic classes and Invariants of Spin Geometry July 29, 2018 14 / 25

Main results by the following practical rules: 1 in accordance to u = w 2r, w 2r+1 or u = w n when n is even, define f (u) := p r, δ 2 (Sq 2r w 2r+1 ) or e 2 n; Haibao Duan (CAS) Characteristic classes and Invariants of Spin Geometry July 29, 2018 15 / 25

Main results by the following practical rules: 1 in accordance to u = w 2r, w 2r+1 or u = w n when n is even, define f (u) := p r, δ 2 (Sq 2r w 2r+1 ) or e 2 n; 2 f (u) := f (w i1 ) f (w ik ) if u = w i1 w ik ; Haibao Duan (CAS) Characteristic classes and Invariants of Spin Geometry July 29, 2018 15 / 25

Main results by the following practical rules: 1 in accordance to u = w 2r, w 2r+1 or u = w n when n is even, define f (u) := p r, δ 2 (Sq 2r w 2r+1 ) or e 2 n; 2 f (u) := f (w i1 ) f (w ik ) if u = w i1 w ik ; 3 f (u) := f (u 1 ) + + f (u k ) if u = u 1 + + u k with u i s distinct monomials in w 2,, w n. Haibao Duan (CAS) Characteristic classes and Invariants of Spin Geometry July 29, 2018 15 / 25

Main results by the following practical rules: 1 in accordance to u = w 2r, w 2r+1 or u = w n when n is even, define f (u) := p r, δ 2 (Sq 2r w 2r+1 ) or e 2 n; 2 f (u) := f (w i1 ) f (w ik ) if u = w i1 w ik ; 3 f (u) := f (u 1 ) + + f (u k ) if u = u 1 + + u k with u i s distinct monomials in w 2,, w n. Based on Theorem 1 it can be shown that Theorem 2 The pair (f, γ) of operations satisfies the following properties: for any u H 2r (B SO(n) ; Z 2 ) one has i) B(u) = ρ 4 (f (u)) + θ(γ(u)); ii) Sq 1 (γ(u)) = Sq 2r Sq 1 (u) + u Sq 1 (u). Haibao Duan (CAS) Characteristic classes and Invariants of Spin Geometry July 29, 2018 15 / 25

Main results Example Take u = w 2r. Then f (w 2r ) = p r by the definition of f. Solving the equation ii) by the coefficients comparison method gives γ(w 2r ) = w 4r + w 2 w 4r 2 + + w 2r 2 w 2r+2. Haibao Duan (CAS) Characteristic classes and Invariants of Spin Geometry July 29, 2018 16 / 25

Main results Example Take u = w 2r. Then f (w 2r ) = p r by the definition of f. Solving the equation ii) by the coefficients comparison method gives γ(w 2r ) = w 4r + w 2 w 4r 2 + + w 2r 2 w 2r+2. Substituting these into the formula i) of Theorem 2 yields that B(w 2r ) = ρ 4 (p r ) + θ(w 4r + w 2 w 4r 2 + + w 2r 2 w 2r+2 ). This formula was first obtained by W.T.Wu (On th Pontryagin classes I,II, III, Acta. Sinica, 1953-54) by computing with the cochain complex associated to the Schubert cells decomposition on B SO(n). Haibao Duan (CAS) Characteristic classes and Invariants of Spin Geometry July 29, 2018 16 / 25

Main results Example Take u = w 2r. Then f (w 2r ) = p r by the definition of f. Solving the equation ii) by the coefficients comparison method gives γ(w 2r ) = w 4r + w 2 w 4r 2 + + w 2r 2 w 2r+2. Substituting these into the formula i) of Theorem 2 yields that B(w 2r ) = ρ 4 (p r ) + θ(w 4r + w 2 w 4r 2 + + w 2r 2 w 2r+2 ). This formula was first obtained by W.T.Wu (On th Pontryagin classes I,II, III, Acta. Sinica, 1953-54) by computing with the cochain complex associated to the Schubert cells decomposition on B SO(n). S.S. Chern suggested a different approach to the formula, which was implemented by Thomas (Trans. AMS, 1960). Haibao Duan (CAS) Characteristic classes and Invariants of Spin Geometry July 29, 2018 16 / 25

Main results For each n 8 we set h(n) = [ ] n 1 2 and let {w2, w (1) 2,, w (h(n) 1) 2 } be the first h(n) terms of the derived sequence of w 2. Haibao Duan (CAS) Characteristic classes and Invariants of Spin Geometry July 29, 2018 17 / 25

Main results For each n 8 we set h(n) = [ ] n 1 2 and let {w2, w (1) 2,, w (h(n) 1) 2 } be the first h(n) terms of the derived sequence of w 2. Applying the operator f of Theorem 2 we get the sequence of integral cohomology classes: {f (w 2 ), f (w (1) (h(n) 1) 2 ),, f (w 2 )} H (B SO(n) ). In view of the fibration CP i B Spin c (n) π B SO(n) we can show Theorem 3 There is a unique set {q, q r, 1 r h(n) 1} of integral cohomology classes on B Spin c (n), degq r = 2 r+1, that satisfies the following system: Haibao Duan (CAS) Characteristic classes and Invariants of Spin Geometry July 29, 2018 17 / 25

Main results For each n 8 we set h(n) = [ ] n 1 2 and let {w2, w (1) 2,, w (h(n) 1) 2 } be the first h(n) terms of the derived sequence of w 2. Applying the operator f of Theorem 2 we get the sequence of integral cohomology classes: {f (w 2 ), f (w (1) (h(n) 1) 2 ),, f (w 2 )} H (B SO(n) ). In view of the fibration CP i B Spin c (n) π B SO(n) we can show Theorem 3 There is a unique set {q, q r, 1 r h(n) 1} of integral cohomology classes on B Spin c (n), degq r = 2 r+1, that satisfies the following system: 1 ρ 2 (q) = π w 2, ρ 2 (q r ) = π w (r) 2 ; 2 2q 1 q 2 = π p 1, 2q r+1 q 2 r = π f (w (r) 2 ). Haibao Duan (CAS) Characteristic classes and Invariants of Spin Geometry July 29, 2018 17 / 25

Main results Regarding H (B Spin c (n)) as a module over its subring π H (B SO(n) ) we have Theorem 4 The cohomology ring H (B Spin c (n)) has the presentation H (B Spin c (n)) = π H (B SO(n) ) Z[δ] (q, q 1,, q h(n) 1 ) that is subject to the following relations 2q 1 q 2 = π p 1, 2q r+1 q 2 r = π f (w (r) 2 ), 4δ q2 h(n) 1 = h. where (q, q 1,, q h(n) 1 ) denotes the free Z module in the simple system q, q 1,, q h(n) 1 of generators (in Borel s notation); δ is the Euler class of the complex spin representation Spin c (n) U(2 h(n) ) (Atiyah and Bott, 1964). Haibao Duan (CAS) Characteristic classes and Invariants of Spin Geometry July 29, 2018 18 / 25

Main results Computing with the Gysin sequence of the circle fibration U(1) B Spin(n) B Spin c (n) CP shows the following result, where π : B Spin(n) B SO(n) is induced by the covering Spin(n) SO(n). Theorem 5 The cohomology H (B Spin(n) ) has the presentation H (B Spin(n) ) = π H (B SO(n) ) (q 1,, q h(n) 1, δ ± ), subject to the following relations: 2q 1 = p 1, 2q r+1 q 2 r = π f (w (r) 2 ),. Haibao Duan (CAS) Characteristic classes and Invariants of Spin Geometry July 29, 2018 19 / 25

Main results Computing with the Gysin sequence of the circle fibration U(1) B Spin(n) B Spin c (n) CP shows the following result, where π : B Spin(n) B SO(n) is induced by the covering Spin(n) SO(n). Theorem 5 The cohomology H (B Spin(n) ) has the presentation H (B Spin(n) ) = π H (B SO(n) ) (q 1,, q h(n) 1, δ ± ), subject to the following relations: 2q 1 = p 1, 2q r+1 q 2 r = 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 Haibao Duan (CAS) Characteristic classes and Invariants of Spin Geometry July 29, 2018 19 / 25

Applications Haibao Duan (CAS) Characteristic classes and Invariants of Spin Geometry July 29, 2018 20 / 25

Applications 1 The spin characteristic classes are more subtle that the Pontyagin classes: there exist spin vector bundles ξ for which p i (ξ) = 0, w i (ξ) = 0, i 1, but q r (ξ) 0 for some r 1 (Thomas, 1962); 2 In the Milnor s calculation (1957) on the group Θ 7 if one uses the Spin characteristic classes q 1, q 2 in place of the Pontryagin classes p 1, p 2, one obtains Θ 7 14 instead of Θ 7 7; 3 For an 8 dimensional manifold M, a pair of integral cohomology classes (a, b) can be realized as the first two spin characteristic classes of a stable spin bundle on M, if and only if where U 1 3 a 2 + b U 1 3 a on H 8 (M; Z 3 ) is the mod3 Wu class of M (Duan, 1991). Haibao Duan (CAS) Characteristic classes and Invariants of Spin Geometry July 29, 2018 20 / 25

Applications 1 The spin characteristic classes are more subtle that the Pontyagin classes: there exist spin vector bundles ξ for which p i (ξ) = 0, w i (ξ) = 0, i 1, but q r (ξ) 0 for some r 1 (Thomas, 1962); 2 In the Milnor s calculation (1957) on the group Θ 7 if one uses the Spin characteristic classes q 1, q 2 in place of the Pontryagin classes p 1, p 2, one obtains Θ 7 14 instead of Θ 7 7; 3 For an 8 dimensional manifold M, a pair of integral cohomology classes (a, b) can be realized as the first two spin characteristic classes of a stable spin bundle on M, if and only if a 2 + b U 1 3 a on H 8 (M; Z 3 ) where U3 1 is the mod3 Wu class of M (Duan, 1991). For such realization problem in the unstable instances, there should be 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 = W Spin(n) invariants c k = i (q k ) H (BT ) W = Z[t 1,, t m ] W, 1 k h(n). Haibao Duan (CAS) Characteristic classes and Invariants of Spin Geometry July 29, 2018 21 / 25

Applications 1 There is a simple recurrence to produced the basic W = W Spin(n) invariants c k = i (q k ) H (BT ) W = Z[t 1,, t m ] W, 1 k h(n). 2 The relations on the cohomology H (B Spin(n) ) 2q r+1 q 2 r = π f (w (r) 2 ) indicates that the ring H (BT ) W of integral W = W Spin(n) invariants is not a polynomial ring. Haibao Duan (CAS) Characteristic classes and Invariants of Spin Geometry July 29, 2018 21 / 25

The proof of main results We conclude with a proof of the main result. Theorem 1. Let X be a δ 2 formal space. There exists a unique pair of cohomological operations F : H 2r (X ; Z 2 ) S 4r 2 (X ; Z 2) H 4r (X ; Z 4 ), written F (u) = (γ(u), α(u)), that satisfies the following properties: i) α(u) Im ρ 4 ; ii) B(u) = α(u) + θ(γ(u)); iii) Sq 1 (γ(u)) = Sq 2r Sq 1 (u) + u Sq 1 (u). Haibao Duan (CAS) Characteristic classes and Invariants of Spin Geometry July 29, 2018 22 / 25

The proof of main results For any topological space X and a cohomology class u H 2r (X ; Z 2 ) there hold the following universal relations: (4.1) δ 2 (u u) = 2δ 4 (B(u)) in H 4n+1 (X ); (4.2) ρ 2 δ 4 B(u) = Sq 2n Sq 1 u + u Sq 1 u in H 4n+1 (X ; Z 2 ). Assume that the space X is δ 2 formal. Then δ 4 (B(u)) Im δ 2 by (4.1). In view of the isomorphism δ 2 : S 4r 2 (X ) = Im δ 2 there exists a unique element u 1 S2 4r (X ) so that (4.3) δ 2 (u 1 ) = δ 4 (B(u)). Haibao Duan (CAS) Characteristic classes and Invariants of Spin Geometry July 29, 2018 23 / 25

The proof of main results For any topological space X and a cohomology class u H 2r (X ; Z 2 ) there hold the following universal relations: (4.1) δ 2 (u u) = 2δ 4 (B(u)) in H 4n+1 (X ); (4.2) ρ 2 δ 4 B(u) = Sq 2n Sq 1 u + u Sq 1 u in H 4n+1 (X ; Z 2 ). Assume that the space X is δ 2 formal. Then δ 4 (B(u)) Im δ 2 by (4.1). In view of the isomorphism δ 2 : S 4r 2 (X ) = Im δ 2 there exists a unique element u 1 S2 4r (X ) so that (4.3) δ 2 (u 1 ) = δ 4 (B(u)). We can now formulate the desired operations F = (γ, α) : H 2r (X ; Z 2 ) S 4r 2 (X ) H4r (X ; Z 4 ) by setting (4.4) γ(u) := u 1, α(u) := B(u) θ(γ(u)). Haibao Duan (CAS) Characteristic classes and Invariants of Spin Geometry July 29, 2018 23 / 25

The proof of main results Applying ρ 2 to both sides of (4.3) we get by (4.2) that Sq 1 γ(u) = Sq 2n Sq 1 u + u Sq 1 u. Haibao Duan (CAS) Characteristic classes and Invariants of Spin Geometry July 29, 2018 24 / 25

The proof of main results Applying ρ 2 to both sides of (4.3) we get by (4.2) that Sq 1 γ(u) = Sq 2n Sq 1 u + u Sq 1 u. Moreover, from δ 4 α(u) = δ 4 (B(u) θ(γ(u))) = δ 4 (B(u)) δ 2 (γ(u)) (by δ 4 θ = δ 2 ) = 0 (by (4.3)) we find that α(u) Im ρ 4. Summarizing, we have obtained the operation F that satisfies the properties i), ii) and iii) of Theorem 1, whose uniqueness comes from its definition (4.4). Haibao Duan (CAS) Characteristic classes and Invariants of Spin Geometry July 29, 2018 24 / 25

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