Intersection index (in homology)

A homology invariant characterizing the algebraic (i.e. including orientation) number of points in the intersection of two subsets of complementary dimensions (in general position) in a Euclidean space or in an oriented manifold. In the case of a non-oriented manifold, the coefficient ring $ R $ for the homology is taken to be $ \mathbf Z _ {2} $.

Let $ X \supset A $, $ Y \supset B $ be pairs of subsets in the Euclidean space $ \mathbf R ^ {n} $ such that $ A \cap Y = \emptyset = X \cap B $, and let $ d: ( X \times Y, ( A \times Y) \cup ( X \times B)) \rightarrow ( \mathbf R ^ {n} , \mathbf R ^ {n} \setminus 0) $ be the mapping given by $ d( x, y) = x- y $. The intersection index $ \xi \circ \eta $ for the homology classes $ \xi \in H _ {n-} i ( X, A) $, $ \eta \in H _ {i} ( Y, B) $ is the element $ (- 1) ^ {i} d _ \star ( \xi \times \eta ) $. Here $ d _ \star $ is the induced homology mapping, while $ \xi \times \eta \in H _ {n} ( X \times Y, ( A \times Y) \cup ( X \cup B)) $ is the exterior homology product of the elements $ \xi $ and $ \eta $.

The intersection index $ \xi \circ \eta $ depends only on those parts of the classes $ \xi $ and $ \eta $ with supports in an arbitrary small neighbourhood $ V $ of the closure of the set $ X \cap Y $. In particular, $ \xi \circ \eta = 0 $ if $ X \cap Y = \emptyset $. Also, if $ V = \cup _ {i} V _ {i} $, $ V _ {i} \cap V _ {j} = \emptyset $ for $ i \neq j $, then the local intersection indices of $ \xi $ and $ \eta $ corresponding to each open set $ V _ {i} $ are defined, and their sum coincides with $ \xi \circ \eta $. The invariant $ \xi \circ \eta $ does not change under homeomorphisms of $ \mathbf R ^ {n} $. In conjunction with the previous property of locality, this enables one to determine the intersection index $ \xi \circ \eta $ for compact subsets of an oriented variety. The following anti-commutative relation holds:

$$ \xi \circ \eta = (- 1) ^ {i(} n- i) \eta \circ \xi . $$

If $ X $ and $ Y $ are vector subspaces in general position, if $ A = X\setminus 0 $, $ B= Y\setminus 0 $, and if $ \xi $ and $ \eta $ are generators of $ R = H _ {n-} i ( X, A) = H _ {i} ( Y, B) $, then $ \xi \circ \eta $ is a generator of $ H _ {n} ( \mathbf R ^ {n} , \mathbf R ^ {n} \setminus 0) = R $. Since the choice of these generators is equivalent to the choice of an orientation in the corresponding Euclidean spaces, this makes it possible to determine the intersection index $ c \circ c ^ \prime $ for two chains of complementary dimensions (including singular ones) for which $ | c | \cap | \partial c ^ \prime | = \emptyset = | c ^ \prime | \cap | \partial c | $( $ | c | $ is the support of the chain $ c $, the boundary of which is $ \partial c $). Then $ c \circ c ^ \prime = \xi \circ \eta $ for certain chains $ c $ and $ c ^ \prime $ of the homology classes $ \xi \in H _ {n-} i ( X, A) $, $ \eta \in H _ {i} ( Y, B) $, $ | c | \subset X $, $ | \partial c | \subset A $, $ | c ^ \prime | \subset Y $, $ | \partial c ^ \prime | \subset B $.

The intersection index is used to describe certain duality relations in manifolds.

References

[1]	A. Dold, "Lectures on algebraic topology" , Springer (1980)

References

[a1]	J.F. Adams, "Stable homotopy and generalised homology" , Univ. Chicago Press (1974)