## 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

**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 *K*-theory, yielding the presentations

where *n* copies of

### A.1 Ring presentations

Let *R* be a unital, commutative ring of characteristic zero, additively generated by elements

By eliminating generators if necessary, we may assume that the only relations *i* (where *R*.
If *R* has characteristic

More generally, say *R* as a ring (but not necessarily as an abelian group), and define

form an additive generating set for *R* (where the

fixing

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

Let Σ be a closed connected surface.
Classes in the ungraded cohomology ring *W* of real vector bundles over Σ takes values in

given by *W* is a homomorphism.
Restricting *W* to

### Lemma A.1.

*The map W is an isomorphism of abelian groups
*

### Proof.

Consider the cofiber sequence *K*-theory has the form

which implies that *W* is surjective.
For every *L* such that

### A.3 The abelian group structure of H * ( Σ ; 𝔽 2 ) ×

There is a general procedure for computing the

One obtains the presentation

where

### Lemma A.2.

*For every *

Let *n* copies of

where

### Lemma A.3.

*For every *

### Proof.

A computation shows that the *k* is *odd* (and *k* is odd, all elements of the form
*even*, the elements

### A.4 The ring structure of K O ( Σ )

We study the products in

*Case *
All products are trivial since

*Case *
Let

*W*of the units

*Case *
For

*W*of the unit

## 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.

## References

[1] A. Adem, F. R. Cohen and E. Torres Giese, Commuting elements, simplicial spaces and filtrations of classifying spaces, Math. Proc. Cambridge Philos. Soc. 152 (2012), no. 1, 91–114. 10.1017/S0305004111000570Search in Google Scholar

[2] A. Adem and J. M. Gómez, A classifying space for commutativity in Lie groups, Algebr. Geom. Topol. 15 (2015), 493–535. 10.2140/agt.2015.15.493Search in Google Scholar

[3] A. Adem, J. M. Gómez, J. Lind and U. Tillman, Infinite loop spaces and nilpotent K-theory, Algebr. Geom. Topol. 17 (2017), 869–893. 10.2140/agt.2017.17.869Search in Google Scholar

[4] O. Antolín-Camarena, S. P. Gritschacher and B. Villarreal, Classifying spaces for commutativity in low-dimensional Lie groups, Math. Proc. Cambridge Philos. Soc., to appear; preprint (2018), https://arxiv.org/abs/1802.03632. 10.1017/S0305004119000240Search in Google Scholar

[5]
O. Antolín-Camarena and B. Villarreal,
Nilpotent *n*-tuples in

[6] J. F. Davis and P. Kirk, Lecture Notes in Algebraic Topology, Grad. Stud. Math. 35, American Mathematical Society, Providence, 2001. 10.1090/gsm/035Search in Google Scholar

[7]
D. Dugger and D. C. Isaksen,
Topological hypercovers and

[8] J. L. Dupont, Curvature and Characteristic Classes, Lecture Notes in Math. 640, Springer, Berlin, 1978. 10.1007/BFb0065364Search in Google Scholar

[9] J. Ebert and O. Randall-Williams, Semi-simplicial spaces, preprint (2018), https://arxiv.org/abs/1705.03774; to appear in Algebr. Geom. Topol. Search in Google Scholar

[10]
S. P. Gritschacher,
Commutative *K*-theory,
PhD thesis, University of Oxford, 2017.
Search in Google Scholar

[11] G. H. Rojo, On the space of commuting orthogonal matrices, J. Group Theory 17 (2014), no. 2, 291–316. 10.1515/jgt-2013-0038Search in Google Scholar

[12] G. Segal, Categories and cohomology theories, Topology 13 (1974), 293–312. 10.1016/0040-9383(74)90022-6Search in Google Scholar

[13] E. Torres Giese and D. Sjerve, Fundamental groups of commuting elements in Lie groups, Bull. Lond. Math. Soc. 40 (2008), no. 1, 65–76. 10.1112/blms/bdm094Search in Google Scholar

**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