It’s time to get our hands dirty with some topology! Instead of studying spaces directly, we’re going to study a way of building topological spaces. In particular, we’re going to view a space as a collection of analogous subspaces appropriately glued together satisfying a few restrictions.
This viewpoint is of large interest in algebraic topology and spaces that can be viewed in this way are called -complexes. So, what is a -complex? One might instead start with a different question: what is the in a -complex. Quite plainly, the stands for triangle. A -complex on a topological space is a way of looking at as being built from triangles and their higher dimensional analogues along with a notion of gluing these triangles along their boundaries. In the following were going to look at -complexes from a formal standpoint and follow up with examples of -complex structures on the torus, Klein bottle, and real projective plane.
A -complex is a space built out of -simplicies and the gluing maps associates with them. An -simplex can be thought of as the -dimensional triangle with an ordering. Precisely, an -simplex is the smallest convex set in containing the points (which we call vertices) such that all the points do not lie in an -dimensional hyperplane and such that the vertices are totally ordered say . The ordering is often written as and we will use this notation from now on. The hyperplane condition is necessary so that there is not a collapsing of the -simplex. In other words, it assures us that is not embeddable in for . The ordering of the vertices is necessary for some technical aspects of the theory. Here are examples of an -simplex for :
It is also important to note that the assumed topology on is the induced subspace topology from .
It can be shown that all -simplicies are homeomorphic to the standard -simplex which is the simplex enclosed by the standard basis vectors for and the origin . Precisely, we can describe the standard -simplex as
with ordering . Since we have an ordering on the vertices there is a homeomorphism from the standard -simplex to any other -simplex by sending the basis vectors (and the origin) to the vertices in the other -simplex with the corresponding ordering and then extending linearly (notice that we can write as ). This justifies us calling the standard -simplex (or any other -simplex for that matter) and only working with standard simplicies.
It’s evident from the image above that if we remove any vertex from an -simplex , then the remaining vertices form and -simplex. Such a simplex is called a face of (the reason we say face is clear from the tetrahedron example above). The ordering of the vertices that make up a face is given the induced ordering from the vertices in . In other words, if has ordering and we remove the vertex , then the ordering on the corresponding face is given by . The subspace of all faces of is called the boundary of and is denoted by . The interior of is . We are now ready to state the properties a topological space must satisfy for it to have a -complex structure.
Let be any topological space. is said to have a -complex structure if there is a collection of continuous functions called gluing maps, where depends on , such that the following three properties hold:
- The restriction is injective, and each point is the image of exactly one such restriction.
- If is restricted to a face of , then there is a map such that on this restriction.
- A subset is open if and only if is open in .
Let us take a moment to discuss what these conditions mean before giving some examples. Intuitively, the first condition says that there is no collapsing of the simplex as it is mapped into except possibly along the boundary, and there is no overlap of the interiors of any two distinct -simplicies. The second condition adds a little more as it tells us how the images of the -simplicies and -simplicies interact. Specifically, the second condition tells us that the images of an -simplex and an -simplex can only overlap on the boundary of the -simplex. Moreover, when there is overlap the -simplex must be mapped nicely enough into as to agree with how the corresponding -simplex is mapped into . The third condition is a little more tricky to understand. It tells us that the maps act like quotient maps on their images. By the injectivity in the first condition, this means that subsets look like a subset of given that lies in the interior of the image of some . With this, the second condition tells us that we may build a -complex structure inductively in the following manner: First choose a set of distinct vertices that is -simplicies (their number determines an upper bound for the dimension of the resulting -complex). Here there are obvious unique maps . Now attach edges, that is -simplicies, between the vertices and to each edge there also corresponds a unique map . Repeat this process for -simplicies and so on. This justifies the earlier terminology of calling the gluing maps.
Let’s give some examples. The image below gives a -complex structures of the torus, Klein bottle, and real projective plane from left to right respectively.
Notice and correspond to the two -simplicies, , , and correspond to the three -simplicies, and and corresponds to the vertices or -simplicies. The orientations on the of the -simplicies are a little more subtle to understand. Generally speaking, if we have an -simplex with vertices in order then there is a natural induced orientation of the edges given by the ordering of the vertices. In other words, the edge , where , is given the orientation from to . Looking at the -complex for the projective plane in the diagram above, this means that . Now you might ask, couldn’t we reverse the orientation of the edge and still end up with a -complex? The answer is no. This is because there were originally four distinct vertices , , , and with the same orientation as given in the figure. But because of identification of the edges, there only ends up being two distinct vertices which we relabel as and . The same subtlety is occurring in the -complexes of the torus and Klein bottle as well. I encourage you to work out -complexes with four distinct vertices for the tours, Klein bottle, and real projective plane and show that after identifications they agree with the ones in the diagram above.
A primary reason that -complexes are useful is in the study of simplicial and singular homology. In fact, simplicial homology is defined only on spaces with a -complex structure while singular homology is defined in terms of continuous maps from -simplicies to the topological space of study. It’s also the case that singular homology has a more geometric interpretation when the spaces of study have -complex structures.
As a final note, -complexes “fit between” two other types of complexes, namely simplicial complexes and CW complexes. What we mean by fit between is that there are more restrictions on the gluing maps for simplicial complexes than delta complexes and more restrictions on delta complexes than CW complexes. The gluing maps for CW complexes can be any continuous map while the gluing maps for simplicial complexes are quite restrictive. As for -complexes, they must satisfy property 2 but they are also required to map different faces of to different -simplicies when we view the restricted map as mapping into a simplex in (we can do this by property 1). For instance, none of the -complexes pictured above are simplicial complexes because the map from to each vertex is the restriction of the corresponding face of each adjacent edge. However all three -complexes are CW complexes.
Algebraic Topology – Allen Hatcher