Homology Theory - Axiomatics and Generalised Homology

Axiomatics and Generalised Homology

There are various ways to define cohomology groups (for example singular cohomology, Čech cohomology, Alexander–Spanier cohomology or Sheaf cohomology). These give different answers for some exotic spaces, but there is a large class of spaces on which they all agree. This is most easily understood axiomatically: there is a list of properties known as the Eilenberg–Steenrod axioms, and any two constructions that share those properties will agree at least on all finite CW complexes, for example.

One of the axioms is the so-called dimension axiom: if is a single point, then for all, and . We can generalise slightly by allowing an arbitrary abelian group in dimension zero, but still insisting that the groups in nonzero dimension are trivial. It turns out that there is again an essentially unique system of groups satisfying these axioms, which are denoted by . In the common case where each group is isomorphic to for some, we just have . In general, the relationship between and is only a little more complicated, and is again controlled by the Universal coefficient theorem.

More significantly, we can drop the dimension axiom altogether. There are a number of different ways to define groups satisfying all the other axioms, including the following:

• The stable homotopy groups
• Various different flavours of cobordism groups:, and so on. The last of these (known as complex cobordism) is especially important, because of the link with formal group theory via a theorem of Daniel Quillen.
• Various different flavours of K-theory: (real periodic K-theory), (real connective), (complex periodic), (complex connective) and so on.
• Brown–Peterson homology, Morava K-theory, Morava E-theory, and other theories defined using the algebra of formal groups.
• Various flavours of elliptic homology

These are called generalised homology theories; they carry much richer information than ordinary homology, but are often harder to compute. Their study is tightly linked (via the Brown representability theorem) to stable homotopy.