Homology, Chain Maps acting on Homology Groups

Chain Maps acting on Homology Groups

Let A and B be chains of modules, with their associated homology groups. Let f be a chain map, i.e. a homomorphism from A into B that commutes with the boundary operator. As we shall see, f induces a map from the homology of A into the homology of B.

Let x lie in the kernel of A7. Travel down to A8 and over to B8 and find 0. Thus f7(x) winds up in the kernel of B7. In other words, f maps kernels to kernels.

Let x lie in the image of y in A6. Starting at y, move down and across, or across and down, and f6(x) lies in the image of B6. In other words, f maps images to images.

Take one more step and map the homology group of A7 into the homology group of B7. Since the image of A7, which is the kernel of our homology group, winds up in the image of B7, the induced map from one homology group into another is well defined. In other words, f induces a map on homology groups.

Composition

Let f map A into B and let g map B into C. The composition of f and g defines a chain map from A into C, and the induced map on homology groups is the composition of the maps induced by f and by g. Either way, a member of the homology group of A7, represented by x, maps to the member of the homology group of C7 represented by g(f(x)).

Sum

Let f and g be two chain maps from A into B. Verify that the sum f+g is another chain map, respecting the internal homomorphisms of the two chain complexes. The sum induces a map on homologies that is the sum of the individual maps. Either way, a member of the homology group of A7, represented by x, maps to the member of the homology group of B7 represented by f(x)+g(x).

Subchain

Let A be a subchain of B. In other words, each Ai is a subgroup (or submodule) of Bi. Let f perform the embedding, whence f induces a map on homology groups.

Although A is a restriction of B, its homology groups could be larger. Let B1 through B5 be the short exact sequence 0 → Z4 → Z8 → Z2 → 0. Let Ai = Bi, except for A2, which is restricted to Z2. The homology group of B3 is trivial, while the homology group of A3 is Z2.

Let g be a chain map from B into C. By composition, the action of g on A produces a map on homologies that is the map induced by embedding, followed by the map induced by g on B. However, we usually don't care about the homology of the subchain. Rather, we are interested in the homology of B, and the associated map into the homology of C, when B is restricted to A. Restricting to A extracts a subgroup of B, and of the homology of B. This in turn maps into the homology of C in the usual way. Thus, restricting a chain of modules restricts the homology groups on those modules, along with the induced homology homomorphisms implied by subsequent chain maps.

Join and Direct Sum

Let Q and R be subchains of B, and let f be a chain map from B into C. Restricting to Q restricts the homology of B to those members represented by some x in Q, and this restricts the homology map from B to C accordingly. Similar results hold for R.

Let J be the subchain generated by Q and R. In other words, Ji is the submodule of Bi spanned by Qi and Ri. The same relationship holds if we mod out by the image of Bi-1. Therefore restricting to J produces the join of the homology groups produced by the restrictions to Q and R, and similarly for the map from the homology of B into the homology of C.

If J is the direct sum of Q and R, then this relationship carries through the homology of B, and the map into the homology of C.