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Commutative cocycles and stable bundles over surfaces

  • Daniel A. Ramras ORCID logo EMAIL logo and Bernardo Villarreal ORCID logo
From the journal Forum Mathematicum

Abstract

Commutative K-theory, a cohomology theory built from spaces of commuting matrices, has been explored in recent work of Adem, Gómez, Gritschacher, Lind and Tillman. In this article, we use unstable methods to construct explicit representatives for the real commutative K-theory classes on surfaces. These classes arise from commutative O(2)-valued cocycles and are analyzed via the point-wise inversion operation on commutative cocycles.

MSC 2010: 55N15; 55R35

Communicated by Frederick R. Cohen


Funding source: Simons Foundation

Award Identifier / Grant number: 279007

Award Identifier / Grant number: 579789

Funding statement: D. Ramras was partially supported by the Simons Foundation (Collaboration Grants #279007 and #579789).

A Real topological K-theory of surfaces

We now describe the relationship between the 𝔽2-cohomology ring of a closed connected surface and its real topological K-theory, yielding the presentations

KO(S2)[e1]/(2e1,e12),
KO(Σg)[lai,lbj:1i,jg]/(2lai,2lbj,lailaj,lbilbj,lailbi+lajlbj,lailbk:ik),
KO(Pn)[lai:1in]/(4lai,lai2-2laj,lailak:ik),

where Σg is the connected, orientable surface of genus g>0 and Pn is the connected sum of n copies of 2.

A.1 Ring presentations

Let R be a unital, commutative ring of characteristic zero, additively generated by elements r1,,rnR. Then there is a surjective ring homomorphism f:[x1,,xn]R sending xi to ri. We have rirj=kakijrk for some akij (i,j{1,,n}), and hence

(A.1)xixj-kakijxkker(f).

By eliminating generators if necessary, we may assume that the only relations kakrk=0 (a1,,an) are of the form airi=0. A simple induction on degree shows that ker(f) is generated by the elements (A.1) together with one element aixi 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 .

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

r1,,rn,p1(r1,,rn),,pk(r1,,rn)

form an additive generating set for R (where the pi are integer polynomials), then we obtain another surjection g:[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

[x1,,xn,p1,,pk][x1,,xn]

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 KO~(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 xiHi(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

W:KO(Σ)H*(Σ;𝔽2)×

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 KO~(Σ) and writing classes in the form E-εnKO~(Σ), where εn is the trivial bundle of rank n=rank(E), we see that W(E-εn)=W(E).

Lemma A.1.

The map W is an isomorphism of abelian groups W:KO~(Σ)H*(Σ;F2)×.

Proof.

Consider the cofiber sequence nS1Σ𝑐S2. The induced exact sequence in K-theory has the form

/2=KO~(S2)KO~(Σ)KO~(nS1)=(/2)n,

which implies that KO~(Σ) has at most 2n+1 elements. Since H*(Σ;𝔽2)× has exactly 2n+1 elements, it suffices to prove W is surjective. For every x1H1(Σ;𝔽2), there is a line bundle L such that W(L)=1+x1. The bundle c*E1 classified by the map Σ𝑐S2g1BSO(2) satisfies w1(c*E1)=0, and w2(c*E1)H2(Σ;𝔽2)𝔽2 is the generator. One now finds that W(c*E1L)=1+x1+w2(c*E1). ∎

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*(M1M2;𝔽2)H*(M1#M2;𝔽2) via the associated Mayer–Vietoris sequence.

One obtains the presentation

H*(Σg;𝔽2)𝔽2[a1,,ag,b1,,bg]/(aiaj,bibj,aibi+ajbj,aibk:ik),

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.

Lemma A.2.

For every g1, there is an isomorphism of abelian groups H*(Σg;F2)×(Z/2)2g+1. Moreover, letting y2H2(Pn;F2) denote the generator, we have the following generating set for H*(Σg;F2)×:

{1+ai,1+bj,1+y2:1i,jg}.

Let Pn denote the connected sum of n copies of 2; note that Pn is non-orientable, and in fact, each closed, connected, non-orientable surface is homeomorphic to Pn for some n1. The graded 𝔽2-cohomology ring of Pn has a presentation

H*(Pn;𝔽2)𝔽2[a1,,an]/(ai3,ai2+aj2,aiak:ik),

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

Lemma A.3.

For every n1, there is an isomorphism of abelian groups H*(Pn;F2)×Z/4×(Z/2)n-1. Moreover, {1+ai:1in} generates H*(Pn;F2)×.

Proof.

A computation shows that the 2n-1 elements 1+ai1++aik, where k is odd (and i1,,ik are distinct), all have (multiplicative) order 4. Furthermore, if y2H2(Pn;𝔽2) denotes the generator, then, when k is odd, all elements of the form 1+ai1++aik+y2 also have order 4, so we have 2(2n-1)=2n elements of order 4 in H*(Pn;𝔽2)×. When k>0 is even, the elements 1+ai1++aik and 1+ai1++aik+y2 (with i1,,ik distinct) have order 2. This yields 2n-1 elements of order 2 in H*(Pn;𝔽2)×. The only abelian group of order 2n+1 with 2n-1 elements of order 2 and 2n elements of order 4 is /4×(/2)n-1. ∎

A.4 The ring structure of KO(Σ)

We study the products in KO~(Σ) using the total Stiefel–Whitney class. We view KO~(Σ) as the kernel of KO(Σ)KO(pt), so that elements in KO~(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 KO~(Σg) denote the inverse images under W of the units 1+ai, 1+bj and 1+y2, respectively (where, as above, y2H2(Σg;𝔽2) is the generator). By Lemmas A.1 and A.2, these classes additively generate KO~(Σg), so the method from Section A.1 yields a presentation of KO~(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 1in, let Lai-ε1KO~(Pn) denote the inverse image under W of the unit 1+ai. Again, these elements form an additive generating set for KO~(Pn) and, after computing the relations, the claimed presentation.

Acknowledgements

We thank Alejandro Adem for encouraging us to examine the question of stability for the classes in Section 4.1, and we thank Simon Gritschacher and Omar Antolín for helpful conversations. Additionally, we thank the referee for many detailed comments that improved the exposition.

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Received: 2018-07-10
Revised: 2019-05-31
Published Online: 2019-07-12
Published in Print: 2019-11-01

© 2019 Walter de Gruyter GmbH, Berlin/Boston

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