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This paper presents a conceptual overview of the main figures that Lacan borrowed in his approach to surface topology; the aim is to fathom the theoretical implications and the ontological consequences of these schemes regarding central psychoanalytic concepts such as subject, jouissance, object, fantasy, among others. It develops a logical sequence of iconic topological figures in Lacan, from the torus to the cross-cap, passing through the Möbius strip and the Klein bottle. It concludes with the cross-cap because it specifies the novel contribution of concepts re-articulated from these new tools. Theoretical terms such as subject and object, as well as their complex relationship, are clarified from a critical conception of dualism and by offering an alternative that surpasses the spheroid topologies. [ABSTRACT FROM AUTHOR]