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On p.98 of the book "Mathematics of the 19th Century: Geometry, Analytic Function Theory", the authors mention a note written by Gauss in 1840:

In 1840 Gauss wrote a note in which he introduced the concept of one- and two-dimensional manifolds, which he called a "line" (Zug) and a "layer" (Schicht), and posed the question of the decomposition of a "layer" bounded by several "lines" into several "layers" bounded by one "line". What is actually involved here is the problem of connectedness of a surface, later solved by Riemann and subsequently became one of the most important problems of combinatorial topology."

My question is mainly about getting references. I tried to search for it in volumes 8, 10 and 12 (i think it's most likely to be found in these volumes). Can anyone help locate it in his writings?

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P.199-206 of Schlesinger's treatise about Gauss's analytic work give the answer. This section, entitled "Anniversary letter Plan of a treatise on convergence of series", deals with mathematical themes that occupied Gauss in the last years of his life (the years 1849 - 1855), and includes a discussion of his last proof of the fundamental theorem of the algebra (published in 1849 and topological in it's nature) as well as a summary of the main ideas in an incomplete treatise on convergence of infinite series. The part of Schlesinger's discussion relevant to this question, is on p. 203-206. Schlesinger mentions that:

Articles 2-5 describe in terms of "layers" delimited by "lines" in the plane of complex variables the basics of topology, whereby it is said with reference to Article 5 of the jubilee that this geometrical interpretation aim ... To facilitate the field of abstract complex sizes and to convey a language for them. A layer can be delimited by one or even several closed lines, and always determines a certain "sequence" in these lines, "similar to when the entire shore lines of a lake including one or more islands are traversed in such a way, that he ... the lake ... either always on the right or always on the left. Art. 5 then shows that "a layer S, which is delimited by two or more closed lines, can be broken down into as many partial layers, each delimited by only one line"; if it is a question of applications, in which two lines, which are identical in themselves but occur in sequential orders, can be viewed as destructive to one another, the totality of the border lines of 2. Put right now, the totality of the simple border lines all of the, , Decomposing layers. It is not difficult to guess which applications Gauss had in mind here...

This translation was made by Google translate with a few corrections i made. I guess that when Gauss speaks about "a lake including one or more islands" he describes a topolgical picture similar to that of a doughnut, and in the next lines he describes the problem of connectedness of a surface. I think this is the reference i searched for.

Although i found the desired reference ,this passage arises new "nuts to crack"; for example, what does Gauss mean when he speaks about "two lines... occur in sequential orders... can be viewed as destructive to one another..."?

Also, it seems that since Gauss speaks of connectivity of surfaces with reference to the complex plane, this treatise can be interpreted as a first step in the creation of geometric foundations for complex function theory (i.e Riemann surface). However, Schlesinger says in p.205 that several things Gauss said confirm that he has not fully penetrated the nature of complex integrals (and multivalued functions).

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