LECTURE 5
5.1. Fary functors: Comments and examples
Our first example of a Fary functor is the ordinary homology F = Z. We can see
that Axioms (1)-( 4) in Section 4.3 correspond to the Eilenberg-Steenrod axioms
for homology. Indeed, Axioms (1) and (2) correspond to functoriality, Axiom (3)
is the long exact sequence, Axiom (4a) corresponds to homotopy, and Axiom (4b)
corresponds to excision.
Axiom ( 4a) is sometimes called constructibility with respect to S. Indeed , it
is the main place in the definition where the stratification S comes in. It does
so through the assumption that a standard pair must b e transverse to S. This
transversality is a codimension 1 condition, so it acquires real force only when we
speak about families of standard pairs.
A comment about the color scheme: if we think of classes in Hi (Y, Z; Z) as
represented by geometric cycles in Y , relative to Z, then the color scheme has the
following meaning. A cycle can have boundary in the green set (green light for a
cycle), but not in the red set (red light).
One may ask why we only define the groups Hi(Y, Z; F) for standard pairs
(Y, Z). The answer is that any pair of compact subsets can be "thickened up" to
a standard pair (cf. Section 4.2), so it is enough just to work with standard pairs,
which are well suited for stating the axioms.
Our second example of a Fary functor is the following. Let 7r : W ___., X be a
proper map which is compatible with S. More precisely, we assume that W has a
stratification such that 7r takes strata of W submersively onto strata of X. This
implies, in particular, that 7r-l ( S) ___., S is a fiber bundle for every stratum S of S.
Then we have a Fary functor F = 7r*Z on X , defined by
(and similarly for the structure maps of F).
Exercise 5.1.1. Check that 7r*Z is a Fary functor.
Example 5.1.2. In the res t of this lecture we will b e using the following two
examples of Fary functors on X = IR^2 = C, stratified by X = {O} U X \ {O}.
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