View source for Intersection index (in homology)

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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|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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Dold,   "Lectures on algebraic topology" , Springer  (1980)</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.F. Adams,   "Stable homotopy and generalised homology" , Univ. Chicago Press  (1974)</TD></TR></table>