Almost Complex Structures on Connected Sums of Complex

Home , Chern class

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Acknowledgements We wish to thank Thomas Friedrich for valuable comments on an earlier version of the paper. We are also grateful to the anonymous referee for his careful reading and helpful comments.

2 Stable almost complex structures on m#CP2n

For a CW complex X let K(X) and KO(X) denote the complex and real K–groups respectively. Moreover we denote by K (X) and KO (X) the reduced groups. Let r : K(X) → KO(X) denote the real reduction map, which can be restricted to a map K (X) → KO (X). We denote the restricted map again withe r. A realg vector bundle F over X has a stable almost complex structure if there is a an element y ∈ K(X) such that r(y)e = F − dimgF . Since r is a group homomorphism, the set of all stable complex vector bundles, such that the underlying real vector bundle is stably isomorphic to Fe, is given by

y + ker r ⊂ K˜ (X), where y is such that r(y) = F − dim F . Let c: KO(X) → K(X) denote the complexiﬁcation map and t: K(X) → K(X) the map which is induced by complex conjugation of complex vector bundles. The maps t and c are ring homomorphisms, but r preserves only the group structure. The following idendities involving the maps r, c and t are well known

c ◦ r =1+ t: K(X) → K(X), r ◦ c = 2: KO(X) → KO(X).

We will writey ¯ = t(y) for an element y ∈ K(X). For two oriented manifolds M and N of same dimension d, we denote by M#N the connected sum of M with N which inherits an orientation from M and N. First, let us characterise the stable tangent bundle of M#N by

Lemma 2.1. Let pM : M#N → M and pN : M#N → N be collapsing maps to each factor of M#N. Then we have ∗ ∗ ∼ d pM (M) ⊕ pN (N) = T (M#N) ⊕ ε .

Proof. Let DM ⊂ M and DN ⊂ N be embedded closed disks and WM and WN collar neigh- borhoods of ∂(M \ D˚M ) and ∂(N \ D˚N ) respectively, where D˚ denotes the interior of D. Thus d−1 d−1 WM =∼ S × [−2, 0] and WN =∼ S × [0, 2]. The manifold M#N is obtained by identifying d−1 d−1 S × 0 ⊂ WM with S × 0 ⊂ WN by the identity map. Set W := WM ∪ WN ⊂ M#N and ∗ ∗ n note that V1 := pM (M) ⊕ pN (N) as well as V2 := T (M#N) ⊕ ε are trivial over W . Moreover d−1 let UM ⊂ M#N be the open set diﬀeomorphic to (M \ WM ) ∪ (S × [−2, −1[) and analogous for UN ⊂ M#N. ∼ ∗ d ∗ Now, since V1|UM = pM (T M)⊕ε and pM (T M)|UM = T (M#N)|UM we have an isomorphism given by ΦM : V2|UM → V1|UM , (ξ, w) 7→ ((pM )∗(ξ), w). For ΦN : V2|UN → V1|UN , we set ΦN (η, w) = (w, −(pN )∗(η)). Moreover both vector bundles V1 and V2 are trivial over W and it is possible to choose trivializations of V1 and V2 over W such that ΦM is given by (v, w) 7→ (v, w) d−1 over WM and such that ΦN is represented by (v, w) 7→ (w, −v) over WN . Over S × [−1, 1] we can interpolate these isomorphisms by

v cos π (t + 1) sin π (t + 1) v 7→ 4 4 w − sin π (t + 1) cos π (t + 1) w 4 4 2 for t ∈ [−1, 1]. Using this interpolation we can glue ΦM and ΦN to a global isomorphism V2 → V1.

Hence T (M#N) − d = T M + TN − 2d in KO(M#N), where T M and TN denote the ∗ ∗ elements in KO(M#N) induced by pM (T M) and pN (TN) respectively. This shows that if M and N admit stable almost complex structures sog does M#N (cf. [5]). For M = N = CP2n we consider theg natural orientation induced by the complex structure of CP2n. We proceed with recalling some basic facts on complex projective spaces. Let H be the tautological line bundle over CPd and let x ∈ H2(CPd; Z) be the generator, such that the total Chern class c(H) is given by 1 + x. The cohomology ring of CPd is isomorphic to Z[x]/hxd+1i. The K and KO theory of CPd are completely understood. Let η := H − 1 ∈ K CPd and CPd ηR := r(η) ∈ KO( ). Then we have e Theorem 2.2g(cf. [11, Theorem 3.9], [2, Lemma 3.5], [8, p. 170] and [13, Proposition 4.3]). (a) K(CPd)= Z[η]/hηd+1i. The following sets of elements are an integral basis of K(CPd)

(i) 1, η, η(η +η ¯),...,η(η +η ¯)n−1, (η +η ¯), ..., (η +η ¯)n, and also, in case d is odd, η2n+1 = η(η +η ¯)n. (ii) 1, η, η(η +η ¯),...,η(η +η ¯)n−1, (η − η¯)(η +η ¯), ..., (η − η¯)(η +η ¯)n−1, and also, in case d is odd, η2n+1

where n is the largest integer ≤ d/2.

d n+1 (b) (i) if d = 2n then KO(CP )= Z[ηR]/hηR i d 2n+1 2n+2 (ii) if d = 4n + 1 then KO(CP )= Z[ηR]/hηR , 2ηR i d 2n+2 (iii) if d = 4n + 3 then KO(CP )= Z[ηR]/hηR i. (c) The complex stable tangent bundle is given by (2n + 1)¯η ∈ K˜ (CP2n) and the real stable tangent bundle is given by r((2n + 1)¯η)) ∈ KO CP2n .

d d (d) The kernel of the real reduction map r : K gCP → KO CP is freely generated by the elements e g d − (i) η − η,¯ (η − η¯)(η +η ¯),..., (η − η¯)(η +η ¯) 2 1, if d is even, (ii) η − η,¯ (η − η¯)(η +η ¯),..., (η − η¯)(η +η ¯)2n−1, 2ηd, if d = 4n + 1, (iii) η − η,¯ (η − η¯)(η +η ¯),..., (η − η¯)(η +η ¯)2n, ηd, if d = 4n + 3.

Next we would like to describe the integer cohomology ring of m#CP2n. For that we introduce the following notation: Let Λ denote either Z or Q. We deﬁne an ideal Rd(X1,...,Xm) in Λ[X1,...,Xm], where X1,...,Xm are indeterminants, as the ideal generated by the following elements

Xi · Xj, i 6= j d d Xi − Xj , i 6= j, d+1 Xj , j = 1,...,m.

Hence we have ∗ d H m#CP ; Λ =∼ Λ[x1,...,xm]/Rd(x1,...,xm) (1)

3 ∗ 2 d 2 d d where xj = pj (x) ∈ H m#CP ; Λ , for x ∈ H (CP ; Λ) deﬁned as above and pj : m#CP → CPd the projection onto the j-th factor. Note that p induces an monomorphism on cohomology. j The stable tangent bundle of m#CP2n in KO(m#CP2n) is represented by

m g (2n + 1) r(¯ηj) j=1 X ∗ 2n 2n 2n where ηj := pj (η) ∈ K CP and r : K(m#CP ) → KO(m#CP ) is the real reduction map. Hence the set of stable almost complex structures on m#CP2n is given by e e m g (2n + 1) η¯j + ker r, (2) j=1 X k ∗ k ∗ −k n−1 n−1 2n For k ∈ N and j = 1,...,m, set wj = pj (H) − pj (H) , ej = ηj(ηj +η ¯j) and ω = η1 .

Proposition 2.3. The kernel of r : K m#CP2n → KO m#CP2n is freely generated by − − − (a) {wk : k = 1,...,n − 1, j = 1,...,m } ∪ {en 1 − en 1 : j = 2,...,m } ∪ {2en 1 − ω}, for n j e 1 gj 1 even,

k (b) {wj : k = 1,...,n, j = 1,...,m}, for n odd. Proof. Consider the cofiber sequence m − i π CP2n 1 −→ m#CP2n −→ S4n. (3) j=1 _ ∗ ∗ Note that the line bundle i pj (H) is the tautological line bundle over the j-th summand of m 2n−1 ∨j=1CP and the trivial bundle on the other summands, since the first Chern classes are the same. For the reduced groups we have m m 2n−1 ∼ 2n−1 K ∨j=1CP = K CP j=1 M ∗ ∗ e e and i pj (η) generates the j-th summand of the above sum according to Theorem 2.2. The long exact sequence in K-theory of the cofibration (3) is given by

−1 m 2n−1 4n 2n m 2n−1 1 4n ···→ K ∨j=1CP → K S → K m#CP → K ∨j=1CP → K S →··· (4) e e −1 e2n−1 e −1 m 2n−1 e From Theorem 2 in [3] we have K CP = 0, hence K ∨j=1CP = 0 and from 1 4n −1 4n Bott periodicity we deduce K S = K S = 0. So we obtain a short exact sequence e e ∗ ∗ 4n π 2n i m 2n−1 0 −→ K S e −→ K me#CP −→ K ∨j=1CP −→ 0. which splits, since the involvinge groupse are ﬁnitely generae ted, torsion free abelian groups. Let 4n ωC be the generator of K S , then the set

∗ k {πe (ωC)}∪ ηj : j = 1,...,m, k = 1,..., 2n − 1 n o 2n 2n ∗ is an integral basis of K m#CP . We claim that ηj = π (ωC) for all j. Indeed, the elements η2n lie in the kernel of i∗, hence there are k ∈ Z such that η2n = k · π∗(ω ). j j j j C e 4 Let ch: K (X) → H (X; Q) denote the Chern character for a ﬁnite CW complex X, then ch is a monomorphism for X = m#CPd (since H˜ ∗(m#CPd; Z) has no torsion, cf. [7]) and an isomorphismf e for X e= Sd onto H∗(Sd; Z) embedded in H∗(Sd; Q). Using the notation of (1) wef have 2n xj 2n 2n e ch(ηj ) = (e − 1) e= xj and using the naturality of ch f

∗ ∗ 2n fch (π (ωC)) = π ch(ωC) = ±xj

∗ since π is an isomorphism onf cohomology inf dimension 2n. We can choose ωC such that ∗ 2n 2n ch(π (ωC)) = xj . This shows kj = 1 for all j and K m#CP is freely generated by f k 2n 2n ηj : j = 1,...,m, k = 1,..., 2n −e 1 ∪ {η1 = · · · = ηm }. n o 2n ∗ ∗ Hence K(m#CP )= Z[η1,...,ηm]/R2n(η1,...,ηm). Since pj (H) ⊗ pj (H) is the trivial bundle we compute the identity −ηj 2 2n ηj = = −ηj + ηj −··· + ηj . 1+ ηj

The ring Z[η1,...,ηm]/R2n(η1,...,ηm) is isomorphic to

m 2n+1 2n 2n Z[ηj]/hη i hη − η : j 6= ii  j  j i j=1 , M   and from Theorem 2.2 the set Γj which contains the elements

n−1 ηj, ηj(ηj + ηj),...,ηj(ηj + ηj) n−1 ηj − ηj, (ηj − ηj)(ηj + ηj),..., (ηj − ηj)(ηj + ηj)

2n+1 2n together with {1} is an integral basis of Z[ηj]/hηj i. Thus the set Γ1 ∪. . .∪Γm ⊂ K(m#CP ) generates the group K(m#CP2n). Observe that e k k−1 k−1 e(ηj +η ¯j) = 2ηj(ηj +η ¯j) − (ηj − η¯j)(ηj +η ¯j) , (5) thus 2n n n−1 n−1 ηj = (ηj +η ¯j) = 2ηj(ηj +η ¯j) − (ηj − η¯j)(ηj +η ¯j) . (6) 2n We set ω := ηj for any j = 1,...,m and

k k ej := ηj(ηj +η ¯j) , j = 1,...,m, k = 0,...,n − 1 k k fj := (ηj − η¯j)(ηj +η ¯j) , j = 1,...,m, k = 0,...,n − 1 and in virtue of relation (6) the set

k k B := {ω} ∪ {ej : j = 1,...,m, k = 0,...,n − 1} ∪ {fj : j = 1,...,m, k = 0,...,n − 2} is an integral basis of K m#CP2n . We proceed with the computation of KO(m#CP2n). We have a long exact sequence for −1 KO-theory like in (4).e From Theorem 2 in [3] we deduce KO CP2n = 0 and therefore g g 5 − − 1 m 2n 1 4n 7 4n 4n+7 KO ∨j=1CP = 0. Moreover KO S = KO S = KO S = 0 by Bott periodicity. Hence we obtain a short exact sequence g g g g 4n 2n m 2n−1 0 −→ KO S −→ KO m#CP −→ KO ∨j=1CP −→ 0. (7) Now we have to distinguishg betweeng the cases where n isg even or odd. We ﬁrst assume that n = 2n−1 n 2l. In that case the ring KO CP is isomorphic to Z[ηR]/hηRi, see Theorem 2.2 (b). Hence CP2n CP2n all groups in (7) are torsion free. Therefore the kernel of r : K m# → KO m# is the same as the kernel of e g ϕ := c ◦ r =1+ t: K m#CP2n → K m#CP2n

2n since r ◦ c = 2 and thus c is a monomorphisme of the torsione free part of KO m#CP . k k k Next we compute a basis of ker ϕ. Using relation (5) we have ϕ(ω) = 2ω, ϕ(ej ) = 2ej − fj k and ϕ(fj ) = 0, thus if g m n−1 k k y = λω + λj ej j=1 =0 X Xk then

m n−1 m m n−2 − ϕ(y) = 2λω + λk(2ek − f k)= 2λ + λn 1 ω + λk(2ek − f k) j j j  j  j j j j=1 =0 j=1 j=1 =0 X Xk X X Xk   n−1 n−1 k k where we used that fj = 2ej − ω by Equation (6). As ω and 2ej − fj , j = 1,...,m, k k = 0,...,n − 2, are linearly independent, we conclude that ϕ(y) = 0 if and only if λj = 0 for j = 1,...,m, k = 1,...,n − 2 and m n−1 λj + 2λ = 0. j=1 X This implies that the set

k n−1 n−1 n−1 {fj : j = 1,...,m, k = 0,...,n − 2} ∪ {e1 − ej : j = 2,...,m} ∪ {2e1 − ω},

n−1 n−1 is an integral basis of ker ϕ. Note that from (6) we have 2e1 − ω = (η1 − η¯1)(η1 +η ¯1) . By an inductive argument we see that

k k+1 1 k (ηj − η¯j)(ηj +η ¯j) = wj + linear combinations of wj , . . . , wj (8) and n−1 n−1 2n−1 2n−1 e1 − ej = η1 − ηj . Thus an integral basis of the kernel, in case n is even, is given by

k n 2n−1 2n−1 {wj : j = 1,...,m, k = 1,...,n − 1} ∪ {w1 } ∪ {η1 − ηj : j = 2,...,m}. Now let us assume that n = 2l + 1. Consider the commutative diagram

∗ ∗ 4n π 2n i m 2n−1 0 K(S ) K m#CP K(∨j=1CP ) 0

rS r# r∨ e e e ∗ ∗ 4n π 2n i m 2n−1 0 KO(S ) KO m#CP KO(∨j=1CP ) 0 g e g 6 8l+4 8l+4 ∗ The map rS : K S → KO S is an isomorphism and therefore i |ker r# : ker r# → ker r∨ is an isomorphism, hence the rank of ker r is mn. We see that the set # e g k n−1 {fj : j = 1,...,m, k = 0,...,n − 2} ∪ {2ej : j = 1,...,m} ∪ {ω}

∗ −1 n−1 2n−1 is an integral basis of (i ) (ker r∨), which follows because ej = ηj − (n − 1)ω and the k structure of the kernel of r∨, see Theorem 2.2 (d) (ii). The elements fj for j = 1,...,m and k = 0,...,n − 2 lie in the kernel of r#. Let

m n−1 n−1 y = λω + λj 2ej j=1 X n−1 n−1 n 2n for λ, λj ∈ Z and suppose r#(y) = 0. From ϕ(ω) = 2ω and ϕ(ej ) = (ηj +η ¯j) = ηj = ω it follows that m n−1 λ + λj = 0. j=1 X k n−1 Hence ker r# is freely generated by the elements fj and 2ej − ω. Observe from (6) that n−1 n−1 2ej − ω = (η − η¯)(η +η ¯) . Thus in case of n odd we deduce like in (8) that the kernel of k r# is freely generated by wj for j = 1,...,m and k = 1,...,n. Hence by Equation (2), stable almost complex structures of m#CP2n for n even are given by elements of the form

m m n−1 m k k n n 2n−1 2n−1 y = (2n + 1) η¯j + aj wj + a1 w1 + bj(η1 − ηj ). (9) i=1 j=1 =1 j=2 X X Xk X and for n odd m m n k k y = (2n + 1) η¯j + aj wj (10) i=1 j=1 =1 X X Xk k for aj , bj ∈ Z. For Theorem 1.2 we have to compute the 2n–th Chern class c2n(E) of a vector 2n−1 2n−1 bundle E representing an element of the form (9) and (10). Let η1 − ηj denote also a 2n 2n−1 2n−1 2n vector bundle over m#CP which represents the element η1 − ηj in K m#CP . The 2n−1 2n−1 total Chern class of η1 − ηj can be computed through the Chern character: We have e 2n−1 2n−1 2n−1 2n−1 2n−1 2n−1 ch(η1 − ηj )= ch(η1) − ch(ηj) = x1 − xj .

The elements off degree k in the Chernf character aref given by νk(c1,...,ck)/k! where νk are the Newton polynomials. The coeﬃcient in front of ck in νk(c1,...,ck) is k (see [9], p. 195) and the other terms are products of Chern classes of lower degree, hence the only non vanishing Chern class is given by 2n−1 2n−1 2n−1 2n−1 c2n−1(η1 − ηj ) = (2n − 2)! (x1 − xj ) Thus the total Chern class of a vector bundle E representing an element of the form (9) is given by

2n+1 c(E) = (1 − (x1 + . . . + xm)) n m m n−1 k a1 aj 1+ nx1 2n−1 2n−1 bj 1+ kxj · (1 + (2n − 2)!(x1 − xj )) 1 − nx1 1 − kxj j=2 j=1 =1 Y Y kY

7 and for (10) m n k aj 2n+1 1+ kxj c(E) = (1 − (x1 + . . . + xm)) , 1 − kxj j=1 k=1 Y Y 2n 2n where the coeﬃcient in front of x1 = . . . = xm is equal to c2n(E). Remark 2.4. Note that for m = 1 (and complex projective spaces of arbitrary dimension) this total Chern class was already computed by Thomas, see [13, p. 130].

3 Almost complex structures on m#CP2n

We now describe an explicit stable almost complex structure on m#CP2n, where m = 2u + 1, for which the assumptions of Theorem 1.2 are satisﬁed, thereby producing an almost complex 2n k structure on m#CP . We choose, in the notation of (9) and (10), aj = 2 for j = 1, . . . , u and k = 1, and all other coeﬃcients 0. Then the top Chern class is as desired:

Proposition 3.1. Let m = 2u + 1 be an odd number. In the cohomology ring of m#CP2n, the 2n 2n coeﬃcient c2n of x1 = · · · = xm of the class

u 2 2n+1 1+ xr c = (1 − (x1 + · · · + x2 +1)) u 1 − x r=1 r Y is 2n c2n = m(2n − 1)+2= χ(m#CP ).

Proof. As xi · xj = 0 for i 6= j, we have

2n+1 2n+1 j0 2n + 1 j0 j0 (1 − (x1 + · · · + x2u+1)) = (−1) (x1 + · · · + x2u+1) j0 j =0 X0 2u+1 2n+1 j0 2n + 1 j0 = (−1) xr . j0 r=1 j =0 X X0 Thus, u 2u+1 2n−1 2 2n+1 c = (1 − xr) (1 + xr) (1 − xs) . r=1 s=u+1 Y Y 2n+1 2n−1 2 The factors (1 − xs) contribute 2n + 1 to c2n, whereas the factors (1 − xr) (1 + xr) contribute 2n − 3. Thus,

2n c2n = u(2n − 3) + (u + 1)(2n + 1) = (2u + 1)(2n − 1)+2= χ((2u + 1)#CP ).

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