Commutative cocycles and stable bundles over surfaces

A.1 Ring presentations

Let R be a unital, commutative ring of characteristic zero, additively generated by elements r1,…,rn∈R. Then there is a surjective ring homomorphism f:ℤ⁢[x1,…,xn]→R sending xi to ri. We have ri⁢rj=∑kaki⁢j⁢rk for some aki⁢j∈ℤ (i,j∈{1,…,n}), and hence

By eliminating generators if necessary, we may assume that the only relations ∑kak⁢rk=0 (a1,…,an∈ℤ) are of the form ai⁢ri=0. A simple induction on degree shows that ker⁡(f) is generated by the elements (A.1) together with one element ai⁢xi for each i (where ai may be zero), yielding a finite presentation of R. If R has characteristic p>0, there is a similar presentation with 𝔽p in place of ℤ.

More generally, say r1,…,rn generate R as a ring (but not necessarily as an abelian group), and define f:ℤ⁢[x1,…,xn]→R as above. If

form an additive generating set for R (where the pi are integer polynomials), then we obtain another surjection g:ℤ⁢[x1,…,xn,p1,…,pk]→R (sending xi to ri and pj to pj⁢(r1,…,rn)), and ker⁡(f) is the image of ker⁡(g) under the map

fixing xi and sending pi to pi⁢(x1,…,xn). By eliminating redundant generators, we may assume that there are no linear relations involving more than one of the generators ri, pj⁢(r1,…,rn). The above procedure then gives a finite generating set for ker⁡(g), and hence for ker⁡(f). The presentations below are obtained this way.

A.2 The total Stiefel–Whitney class and K⁢O~⁢(Σ)

Let Σ be a closed connected surface. Classes in the ungraded cohomology ring H*⁢(Σ;𝔽2) can be uniquely written as x0+x1+x2, where xi∈Hi⁢(Σ;𝔽2) for i=0,1,2. The multiplicative units H*⁢(Σ;𝔽2)× form an abelian group under the cup product, and each unit has the form 1+x1+x2. The total Stiefel–Whitney class W of real vector bundles over Σ takes values in H*⁢(Σ;𝔽2)× and extends to a well-defined map

given by W⁢(E-F)=W⁢(E)⁢W⁢(F)-1. Moreover, since H*⁢(Σ;𝔽2) is a commutative ring, W is a homomorphism. Restricting W to K⁢O~⁢(Σ) and writing classes in the form E-εn∈K⁢O~⁢(Σ), where εn is the trivial bundle of rank n=rank⁡(E), we see that W⁢(E-εn)=W⁢(E).

A.3 The abelian group structure of H*⁢(Σ;𝔽2)×

There is a general procedure for computing the 𝔽2-cohomology ring of a connected sum of surfaces M1⁢#⁢M2 by analyzing the surjection in cohomology H*⁢(M1∨M2;𝔽2)→H*⁢(M1⁢#⁢M2;𝔽2) via the associated Mayer–Vietoris sequence.

One obtains the presentation

where deg⁡(ai)=deg⁡(bi)=1. A computation now shows that every element in H*⁢(Σg;𝔽2)× has order 2, which yields the following result.

Let Pn denote the connected sum of n copies of ℝ⁢ℙ2; note that Pn is non-orientable, and in fact, each closed, connected, non-orientable surface is homeomorphic to Pn for some n≥1. The graded 𝔽2-cohomology ring of Pn has a presentation

where deg⁡(ai)=1. (For n>1, the relation ai3=0 follows from the other relations.)

A.4 The ring structure of K⁢O⁢(Σ)

We study the products in K⁢O~⁢(Σ) using the total Stiefel–Whitney class. We view K⁢O~⁢(Σ) as the kernel of K⁢O⁢(Σ)→K⁢O⁢(pt), so that elements in K⁢O~⁢(Σ) are represented by virtual bundles E-εn with n=dim⁡(E).

Case Σ=S2. All products are trivial since S2 is a suspension.

Case Σ=Σg. Let Lai-ε1, Lbj-ε1 and Ey2-ε2 in K⁢O~⁢(Σg) denote the inverse images under W of the units 1+ai, 1+bj and 1+y2, respectively (where, as above, y2∈H2⁢(Σg;𝔽2) is the generator). By Lemmas A.1 and A.2, these classes additively generate K⁢O~⁢(Σg), so the method from Section A.1 yields a presentation of K⁢O~⁢(Σg) with these elements as generators. After computing the relevant relations, the resulting presentation is readily reduced to that given above.

Case Σ=Pn. For 1≤i≤n, let Lai-ε1∈K⁢O~⁢(Pn) denote the inverse image under W of the unit 1+ai. Again, these elements form an additive generating set for K⁢O~⁢(Pn) and, after computing the relations, the claimed presentation.