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)