Those maps of a closed surface to the three-dimensional torus that are homotopic to embeddings are characterized. Particular attention is paid to the more involved case when the surface is nonorientable.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm199-3-1, author = {Allan L. Edmonds}, title = {Embedded surfaces in the 3-torus}, journal = {Fundamenta Mathematicae}, volume = {201}, year = {2008}, pages = {195-212}, zbl = {1147.57019}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm199-3-1} }
Allan L. Edmonds. Embedded surfaces in the 3-torus. Fundamenta Mathematicae, Tome 201 (2008) pp. 195-212. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm199-3-1/