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\centerline{\bf Becker-Gottlieb Transfer}
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The Becker-Gottlieb transfer is an S-map from the base
to the total space of a fibration. It thus induces homomorphisms on 
homology and cohomology which are also called Becker-Gottlieb transfers.

Suppose that $E @>p>> B$ is a Hurewicz fibration whose fibre $F$ is homotopy
equivalent to a compact CW complex $F$. The Becker-Gottlieb transfer is an
S-map $\tau:B\to E$, which means that there is a map 
$\tau:\Sigma^NB\to\Sigma^NE$ defined between the 
$N$-th suspensions of $B$ and $E$ for some $N$. 
Thus $\tau$ induces a homomorphism
$\tau_*$ on any homology theory and $\tau^*$ on any cohmology theory.

We obtain striking relations on homology and cohomology respectively:
$$
p_*\circ \tau_*=\chi(F), \ \ \tau^*\circ p^*=\chi(F)\tag1
$$
 where $\chi(F)$
denotes multiplication by the Euler-Poincare number of $F$. 


The Becker-Gottlieb transfer was discovered in the mid seventies. It 
generalizes the transfer for finite
covering spaces ($F$ is a finite set of points), which had been well-known
since the forties and which was a generalization of a group theory transfer
from the twenties.

Since the discovery of the Becker-Gottlieb transfer, other transfers have
been discovered which satisfy equations (1) with the Euler-Poincare
number replaced by another elementary topological invariant.  Thus 
$$
p_*\circ \tau_*=k, \ \ \tau^*\circ p^*=k  \tag2
$$
where $k$ denotes multiplication by an integer $k$, which
can be a Lefschetz number, a coincidence number, a fixed point index, a vector
field index, or an intersection number.  For any map $p:X\to Y$, the greatest
common divisor of all the $k$'s associated to a transfer is the Brouwer degree
of $p$ when $X$ and $Y$ are manifolds of the same dimension.  Some of these
transfers are induced by S-maps, and others are not.

Several mathematicians contributed to the discovery of these other transfers,
most of which either generalize some or all of the Becker-Gottlieb transfer.
Albrecht Dold's fixed point transfer was discovered independently of the
Becker-Gottlieb transfer, which it generalizes.
It is sometimes called the Becker-Gottlieb-Dold transfer.

Reference:

James C. Becker and Daniel H. Gottlieb, Vector fields and Transfers,
Manuscripta Mathematica, 72(1991), pp.111-130.
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