The Whitehead theorem in homotopy theory basically says that to fully understand the homotopy type of a CW-complex one only needs to know about the homotopy groups (really, the weak homotopy type). It is very easy to produce spaces to which Whitehead’s theorem doesn’t apply. For instance, consider an infinite wedge of circles and the one-dimensional space pictured below:
There is clearly a continuous bijection , which induces an isomorphism on all homotopy groups. However, and are not homotopy equivalent because the topological fundamental group of , which is a homotopy invariant, is discrete while the topological fundamental group of is not discrete.
Heavier machinery is required to differentiate spaces which are more complicated on a local level than CW-complexes. One of the most traditional approaches to this problem is Shape Theory, initially developed by Carl Borsuk and refined and expanded in the 1970’s by many, including the authors of .
A common technique in mathematics is to approximate complicated objects by simpler ones. For instance, approximating functions in Calculus by Taylor polynomials of increasing degree. This is basically the approach of shape theory: approximate complicated spaces by simpler ones, in particular polyhedra (spaces built out of line segments, triangles, tetrahedra, etc.). Borsuk, the inventor of Shape Theory, first used ANR’s to study the topology of compact metric spaces. The more modern approach pioneered by Segal and Mardesic  is categorical in nature and makes heavy use of inverse systems of polyhedra.
Though Shape Theory helps a great deal in our understanding of complicated spaces; however, it has its limits (pun intended). For instance Shape Theory tells you absolutely nothing about the Griffiths Twin Cone because it is “shape equivalent” to a point!
So where do we start?
Let’s look at the Cech Expansion of a space . This is supposed to let us approximate spaces by simpler ones. We’ll start with an open cover of our space . Even when you forget about the points in each set , the open cover still gives a vague picture of . For instance, take the following cover of the unit circle as a subspace of the plane.
Now forget about the points in the space.
The shape left still closely resembles that of a circle. We can even recover the circle from this data: replace each open set with a point. If two open sets intersect, draw a line segment between the two corresponding points.
We get a polyhedron homeomorphic to the circle. It isn’t usually true that we will get back the original space; this only happens in very special cases and for very fine open covers.
The nerve of an open cover
Definition: An abstract simplicial complex is a set and a set consisting of finite subsets of such that if and , then . A vertex or 0-simplex is a singleton and an n-simplex is a set containing elements. The n-skeleton of is the set
Now if is an open cover of , we construct an abstract simplicial complex called the nerve of . An element of is a finite set such that . The geometric realization is a geometric complex obtained by pasting simplices together using as instructions.
Definition: The geometric realization of an abstract simplicial complex with vertex set is the topological space defined as a subset of the product of functions . In particular, is the set of functions such that
- (in particular all but finitely many are zero),
Give the weak (or induced) topology so that is open in if and only if is open in for all finite sets . is topologized with the subspace topology of .
It is common to write the simplex in spanned by vertices in as .
While is defined as the geometric realization of the nerve, it is a bit more intuitive to think of it in the following way.
Here is the cover:
0 skeleton – A vertex of is a set
1 skeleton – If , then place an edge (1-simplex) between and .
2 skeleton – If , then there are three edges joining each pair of the vertices . Place a triangle (or 2-simplex) so that the edges of the triangle match up with these three edges. In the picture, fill in the each empty triangle with a triangle.
3 skeleton – If , attach a tetrahedron to fill in the boundary that exists from the four triangles.
Input the tetrahedron into the place where it obviously goes (on the right).
In our example, we stop here and leave it embedded in . In general, you would continue to add higher dimensional simplices and give the resulting geometric simplicial complex the weak topology. In addition, the space might not be compact and open covers would typically contain infinitely many sets.
Here’s an observation we can go ahead and make. The order of an open cover of (if it exists) is the smallest number such that every point lies in elements of . Hence, if the order of is , then at most distinct elements can intersect and can only have simplices of dimension .
Proposition: If the order of is , then .
The nerve is supposed to be an “approximation” of the original space . What if it is a bad approximation? You should take a “closer look” at by using a finer open cover of , i.e. one consisting of smaller open sets.
Definition: An open cover is a refinement of another cover if for each there is a such that .
If refines , then is “larger” than since there are more sets in . It makes sense to think of as being a better approximation to since if we collapse the appropriate simplices of , we get back something homotopy equivalent to . This is captured in the next proposition.
Proposition: If is a refinement of another cover , there is a there is an onto simplicial map . The map induced on geometric realizations is unique up to homotopy.
Proof. First define on vertices (i.e. elements of ). If and for , define . If , then clearly so we define on the 1-simplex spanned by and to the 1-simplex spanned by and . Any map defined in this way is called a projection.
The same goes for higher simplices; if , then and we send the simplex to . This gives a well-defined simplicial map on the nerves.
Though it seems like there is a lot of freedom in defining a projection, the choice is independent of homotopy. By construction, any two projections induce contiguous maps on geometric realizations. But contiguous maps of simplicial complexes are homotopic, proving the proposition.
Note that if is a refinement of and is a refinement of , the composition is a canonical map. Thus if is another projection, then there is a homotopy . Now if we let denote the homotopy class of , we have strict equality . Therefore, since open covers of form a directed set , we have an inverse system of homotopy classes of nerves of covers.
Definition: For a paracompact Hausdorff space , the Cech Expansion of is the inverse system in the homotopy category of polyhedra.
Of course, even if is not paracompact Hausdorff, you still get an inverse system; the problem with non-paracompact spaces is that it is much harder to relate to the inverse system without “enough” partitions of unity to build maps .
 S. Mardsic and J. Segal, Shape theory, North-Holland Publishing Company, 1982.