I’m planning on writing a series of posts that explore the theory and applications of characteristic classes in algebraic topology, following the book from Milnor and Stasheff. Here I start with the appendix A, where homology is discussed and basic theorems are laid out. This post will also serve me well as a quick reminder, since I keep forgetting small details and ideas regarding (singular) (co)homology. Many results and explanations can be found in Hatcher’s book.
Singular homology
The standard -simplex is the set consisting of all -tuples with the following two properties:
- for each , we have ;
- and .
The second property says that points in are such that the dot product of the vector with the vector is zero, i.e. lies in the -hyperplane with normal , translated by one unit in any direction. Hence each is an affine space of dimension . For instance:
- is a single point ;
- is the line segment in going up from to ;
- is the (filled) triangle in having as vertices the three standard basis vectors;
- etc.
As a special case, we also define to be the empty set.
In general each has vertices, which are the points in corresponding to the standard basis vectors. Another way to talk about the standard -simplex is to say it is the convex hull of the standard basis vectors in . We can label each vertex with an integer , where is the (zero-based) position of the unique in the standard basis vector corresponding to that vertex.
For each , we have a way to talk about the -th side of a standard -simplex via the function , which is defined as (As usual the hat over a variable in an enumeration means that variable is actually omitted from the enumeration.) Thus the -th side is the convex hull of all vertices of that are not labelled with . Notice that is actually an affine embedding. Moreover, it gives an orientation to each side of a standard simplex: the orientation is positive is even, and negative otherwise.
Again, as a special case, we define the -th side of to be the unique function from the empty set (i.e. ) to the singleton (i.e. ).
Let be any topological space. A singular -simplex in is a continuous map from to . The idea here is to identify such a map with its image; since there are no restrictions on what this image may look like except that nearby points stay nearby (continuity), there could be collapsing or other weird things happening to when viewed through the map. That’s why it’s called a singular simplex.
For , the -th face of a singular -simplex is the singular -simplex given by
For each , the singular chain group with coefficients in a commutative ring is the free -module having one generator for each singular -simplex in . In other words, consists of the formal -linear combinations of -simplices in : Notice that this is a module, even though we use the term group. Just another fun little opportunity to be confused down the line. When , the singular chain group is defined to be the zero module. In the special case , the singular chain group at is the free abelian group on the singular -simplices.
The singular chain group is the algebraic realization of how “sticking triangles on a space” works. If , then glueing two copies of the same simplex one on top of another means they “cancel out”, and when only one of their sides are overlapping, these overlapping sides also cancel out; you obtain a square in the space instead of two triangles. Many geometric arguments use the fact that sides on the boundaries cancel out. The precise meaning of this in is given by the boundary homomorphism, which is a -linear map defined as This definition is made for ; if we simply say . The sign in the formula represents the orientation of each side (recall: the -th side has positive orientation when is even, and negative orientation otherwise). This is done because we need “identical sides” that are “going in opposite directions” to cancel out in many geometrical arguments. For instance, the boundary of a tiling of some region of space by triangles (i.e. a sum in the singular chain group ) should be the boundary of the region, not the sum of the individual boundaries of each triangle (for instance, think about the proof of Stoke’s formula).
An important property of the boundary homomorphism is this: . Intuitively: the boundary of the boundary is empty. Think of , which is a (filled) triangle, and think about it as a singular simplex in (maybe via the projection on the plane in which lies). Its boundary is the formal sum of three line segments. These segments are all perfectly lined up so that the end of the one is the start of the next: they form a cycle. Moreover, the point corresponding to the end of one segment has the opposite orientation to the point corresponding to the start of the next segment, so they cancel out. Since these endpoints are the boundaries of each of the three line segment, and since they all cancel out, the boundary of the boundary is effectively zero.
In fact, the boundary of a singular chain is zero precisely when all of the summands in the chain are arranged so that their boundaries all cancel out, that is, when they form a cycle and “enclose” some region of space. We define the -cycles to be the set of all such chains: The -boundaries is the set of all chains that can be expressed as the boundary of some -dimensional singular chain: The identity says we have a containment of -submodules Hence we can consider the -th singular homology group
The homology group captures in algebra an intuitive spatial fact. We have seen that any cycle “encloses” a region of space, by sticking together simplices of the same dimension along their boundaries until none are “left alone” (each boundary has a matching, opposite, boundary). Now suppose a given cycle is itself the boundary of some higher-dimensional simplex . Now the cycle can “move inside” without breaking appart, shrinking until it becomes a single point. If a cycle is not a boundary, it means that something about the space obstructs the construction of a simplex which would have the cycle as its boundary: there’s a hole in . Notice that the condition of continuity on singular simplices is essential here: the hole would basically force any to be torn appart if it were to have the cycle as a boundary, breaking continuity. In this way, the homology construction detects holes in and gives useful information about them. This technology could be used to make a precise definition of what a “hole” in a topological space is: a hole is a generator for the homology group.
Some abstract nonsense
Homology is a functor from the category of topological spaces up to homotopy, to the category of -modules. The source category’s objects are topological spaces, and the arrows are equivalence classes of continuous maps, where two maps are considered to be the same when they are homotopic. Since I always forget the details of what this means, here they are: consider two continuous maps between topological spaces. We say they are homotopic when there exists another continuous map (called an homotopy) such that and . In the source category we’re interested in, an arrow is a actually a set of continous maps, all homotopic to each other.
Functoriality gives us the following for free: if two spaces are homotopic, then they have isomorphic homology groups. In particular, since is contractible (i.e. homotopy equivalent to a point) for any , we have a concrete homology computation: for any , (It’s easy to compute homology of a point: all singular simplices are the same!) When , things are a bit weird, and fixing weird things is the purpose of the next section.
Reduced homology
There’s a technical point to address here. Consider what happens when we take the space to be a single point and we compute the -th homology group. Since singular -simplices are continuous maps , all -simplices are actually the same. Hence they are all cycles, so by definition their boundaries are zero: . Therefore . However, we set to be the zero map for all earlier, so is , the free module generated by all -simplices. Since they are all the same, there’s actually only one generator, so . For technical reasons, it’s better for the zeroth homology of a point to be the zero module; also, it makes sense intuitively, since we expect the homology to measure holes in a space, and we feel a point doesn’t have holes.
There’s an easy fix to this. Instead of having at degree zero, we set to be (recall: we made a special case above, where the only singular -simplex is the unique function from the empty set to ; then the chain group of degree has to be the free -module generated by that single -simplex), and now we may define at degree zero with the same formula we used for positive degrees. For any singular -simplex , we now have which is the unique function from the empty set to , and which is identified with . Therefore corresponds to the identity map on . This means is trivial, and so is this modified homology at degree zero.
This modified homology is called reduced singular homology and its homology groups are denoted with a tilde: Since the only modification happens at degree zero, we have for each . In general, we see that for any -chain (these are just formal linear combinations of points in ) we have
There is an easy way to get from unreduced homology to reduced homology: at all positive dimensions the groups are the same, and at dimension zero we have the equation (Hint: think about what happens if is path-connected, and which -chains are boundaries.)
Mayer-Vietoris sequence
A great tool for computing with homology. It works for “unreduced” and reduced homology (just replace with everywhere). Let and be two subsets of such that their interior cover (and for reduced homology, we also want their intersection to be nonempty). Then there is a long exact sequence in homology
We can use this to compute the homology of the -sphere . Let be the “open north cap” and be the “open south cap”, i.e. and are contractible open sets in such that their intersection is homotopic to the “equator” . Then the reduced Mayer-Vietoris sequence looks like
Because and are contractible, the middle term above is zero for all , so we have a collection of isomorphisms . Since and zero otherwise, we find by induction on the following calculation:
Relative homology
We now consider pairs where is any subspace of (including the empty subspace, and the full subspace). We’re going to look at homology “modulo ”, in the sense that any singular simplex whose image lies completely in is going to be considered as “completely collapsed”, i.e. zero as a chain. Formally, we define the relative -th singular chain group to be
Because carries chains in to chains in , we obtain a chain complex “modulo ” and we can define relative homology as
Any pair gives an exact sequence of -modules From the general theory of abelian categories, we obtain from it a long exact sequence in homology:
This long exact sequence also exists for reduced homology.
An important tool for working with homology is the excision theorem: let and be two subspaces of such that their interior cover ; then the inclusion induces isomorphisms for all . Equivalently, for any subspaces such that the closure of is contained in the interior of , the obvious inclusion induces isomorphisms This version of the statement is what justifies the name “excision”, since it gives us conditions under which we may excise from without changing the homology groups. That is insanely powerful. For instance, here’s a proof of the so-called Brouwer’s invariance of domain: if and are two nonempty homeomorphic open sets, then . The idea of the proof is to look at what happens locally around a point, so we define local homology groups around some point by Back to the proof. Let be a homeomorphism and pick some point . Then, by excision, we have . (Hint: in the excision theorem, pick to be the complement of and pick to be ). The long exact sequence for the pair looks like: We saw that for every . Hence we have a collection of isomorphisms Moreover, because is homotopic to , the homology is for and zero otherwise. This gives the following calculation: the homology is zero if and only if . The same reasoning applied to and gives the same calculation, and since induces an isomorphism of homology groups, we must have .
Singular cohomology
Since all elements in a -module may be identified with the -linear maps , we may dualize and consider linear maps of the same kind but having opposite polarity: this we do. The -th cochain group is the dual module consisting of all -linear maps going from the singular chain group into its ring of scalars. A cochain is just a way to (linearly) compute a scalar quantity from a singular chain. There’s an analogy to be made with geometry: you have points in some affine space (singular chains), and you have coordinates (cochains), which in a way compute a number from each point. In geometry, there’s a strong link between points and coordinates: studying algebraic varieties is essentially the same as studying rings of coordinates, which are basically rings of polynomial functions from the space to the underlying field. However, it is easier to work with coordinates than sets of points, because there’s a natural ring structure. Hence, if this analogy is to hold, one would expect a link between homology and cohomology; moreover, it should be easier to work with cohomology than homology. And so it is.
The value of a cochain on a chain will be denoted and is defined as Obviously is -bilinear.
The coboundary of a cochain is defined to be the cochain whose value on each -chain is determined by the identity Hence is, up to sign, the dual of , in the sense that for any cochain , the cochain is, up to sign, the pullback of along : This sign convention is used in Milnor and Stasheff’s book, but not in Hatcher for instance, where he defines as precisely the dual of . Since my goal is to understand characteristic classes, I’m going to keep the sign convention used in the M&S book.
Again, this definition has intuitive content: since is able to “measure”, or “give coordinates”, any -chain, then it should be possible to obtain a way to measure -chains by combining measures for the boundaries of .
The coboundary homomorphism, just like its dual friend, verifies . Therefore, if we define -cocycles to be and -coboundaries to be then we may also define the -th singular cohomology group by
Universal coefficient theorem for cohomology
Instead of using the ring as coefficients, we may also use any -module . If is a principal ideal domain, then there is a natural split exact sequence The map is the canonical map sending a cohomology class represented by a cochain , to the map which sends any homology class represented by a chain to the element of .
This exact sequence measures how close the cohomology group is to be the dual of homology.