A correspondence between various Pin-type structures on a compact surface and quadratic (Iinear) forms on its homology is constructed. Sum of structures is defined and expressed in terms of these quadratic forms and in terms of Whitney sum of Spin structures.
DEGTYAREV, A. and FINASHIN, S. (1997) "Pin-Structures on Surfaces and Quadratic Forms," Turkish Journal of Mathematics: Vol. 21: No. 2, Article 7. Available at: https://journals.tubitak.gov.tr/math/vol21/iss2/7