In this work we compute the first integral cohomology of the pure mapping class group of a non-orientable surface of infinite topological type and genus at least 3. To this purpose, we also prove several other results already known for orientable surfaces such as the existence of an Alexander method, the fact that the mapping class group is isomorphic to the automorphism group of the curve graph along with the topological rigidity of the curve graph, and the structure of the pure mapping class group as both a Polish group and a semi-direct product.

The first author was supported during the creation of this article by the research project grants UNAM-PAPIIT IA104620 and UNAM-PAPIIT IN102018. The second author received support from a CONAHCYT Posdoctoral Fellowship and from UNAM-PAPIIT-IN105318. Both authors were supported during the creation of this article by the CONAHCYT Ciencia de Frontera 2019 research project grant CF 217392. Both authors thank Ulises A. Ramos-García for his thoughtful comments on this work. Both authors also thank the referee for their comments and suggestions which helped greatly improve this work.