Topological Manifolds (Pt. II)
Point of Post: This is a continuation of this post.
In the above example, with and just any open subset of we see that the inclusion (where is the inclusion) constitutes an -dimensional continuous atlas for . Thus, open subsets of Euclidean space are themselves topological manifolds.
We claim that the circle is a topological manifold of dimension . Indeed, let be the chart described previously and let be the chart defined by and –one can check, exactly as in the case of , that this is chart. Clearly then and thus constitutes a -dimensional atlas for .
In fact, the -sphere is an -dimensional topological manifold. We basically do the same procedure as the sphere. Namely, we pick two points, and successively exclude those points from the sphere, from where we “unfold” the remains to get all of -space. Indeed, define and . Consider then the sets and and the maps
One can easily verify (using similar logic to the one dimensional case) that is an atlas for .
We now discuss one last example of a topological manifold. We define the real projective space to be the quotient space where if for some . In other words, we can think about as being the space of one-dimensional subspaces of . It is easy to show that the map given by sending to it’s span, is a quotient map. We leave it to the reader to prove that this is second countable and Hausdorff (or see any standard texts on manifolds). To see that is locally Euclidean of dimension . Denote the elements of as (the colons are supposed to make one think of ratios–the thing constant on subspaces). Let and let . This is open because it’s preimage, , is open. Moreover, form an open cover of . Noting then that the map given by
is a well-defined homeomorphism.
Ways to Manufactures Topological Manifolds
There is many a-way to manufacture topological manifolds from old ones, and even sometimes from thin air! Probably the nicest is the following:
Theorem: Let be open and let be a continuous map. Then, if denotes the graph of one has that is a topological manifold of dimension .
Proof: Define by . Clearly this map is continuous, and is bijective. It’s inverse is given by and since each coordinate function is continuous so is the total map. Thus, is an atlas for .
While there is a more snappy result to the above theorem–namely that (for us, will mean homeomorphic)–it is much nicer to have written down an explicit atlas.
This tells us that things like parabolas and sine curves are topological manifolds.
The next obvious way to get manifolds is to take their product and sum.
Let and be topological manifolds of dimensions and respectively. Suppose then that we have atlases and for and . Define, for each , the map as follows: let , then . By definition of the product of topological spaces the map is a homeomorphism . Since are open in and respectively we have that . Moreover, it’s clear that constitutes a cover for . Now, since Hausdorffness and second countability are preserved under finite products we may summarize all of this with the following:
Theorem: Let and be an -dimensional and -dimensional topological manifold respectively. Then, is a topological manifold of dimension . In particular, if and are atlases for and respectively then is an atlas for .
Of course, the theorem above extends to the product of finitely many manifolds So for example the -torus is a topological manifold of dimension . Of course, the -torus is homeomorphic to the surface-of-a-doughnut that we usually call a torus.
The second way we can construct new manifolds out of old ones is via the disjoint union of topological spaces. Indeed, suppose that we have two -dimensional topological manifolds and with atlases and respectively, and let us consider their disjoint union . For each chart on define by . Similarly, define for each chart on . It’s fairly easy to see that and are charts on and their domains cover . Thus, modulo some technical details (which I have previously worked out) we have the following theorem:
Theorem: Let and be two topological manifolds of dimension , then is a topological manifold of dimension . In particular, if and are atlases for and respectively then is an atlas for .
This tells us that things like the disjoint union of two circles is a one dimensional topological manifold.
 Lee, John M. Introduction to Smooth Manifolds. New York: Springer, 2003. Print.
 Lee, John M. Introduction to Topological Manifolds. New York: Springer, 2000. Print.
 Milnor, John Willard, and David W. . Weaver. Topology from the Differentiable Viewpoint. Charlottesville (Va.): University of Virginia, 1969. Print.