Becker-Gottlieb transfer

An $ S $- mapping 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 $ p : E \rightarrow B $ is a Hurewicz fibration (cf. Covering homotopy) whose fibre $ F $ is homotopy equivalent to a compact CW-complex $ F $. The Becker–Gottlieb transfer is an $ S $- mapping $ \tau : B \rightarrow E $, which means that there is a mapping $ \tau : {\Sigma ^ {N} B } \rightarrow {\Sigma ^ {N} E } $ defined between the $ N $ th suspensions of $ B $ and $ E $ for some $ N $( cf. Suspension). Thus, $ \tau $ induces a homomorphism $ \tau _ {*} $ on any homology theory and $ \tau ^ {*} $ on any cohomology theory.

This gives striking relations on homology and cohomology, respectively:

$$ \tag{a1 } p _ {*} \circ \tau _ {*} = \chi ( F ) , \tau ^ {*} \circ p ^ {*} = \chi ( F ) , $$

where $ \chi ( F ) $ denotes multiplication by the Euler–Poincaré number of $ F $.

The Becker–Gottlieb transfer was discovered in the mid-1970s. It generalizes the transfer for finite covering spaces ( $ F $ is a finite set of points), which had been well-known since the 1940s and which was a generalization of a group theory transfer from the 1920s.

Since the discovery of the Becker–Gottlieb transfer, other transfers have been discovered which satisfy equations (a1) with the Euler–Poincaré number replaced by another elementary topological invariant. Thus,

$$ \tag{a2 } p _ {*} \circ \tau _ {*} = k, \quad \tau ^ {*} \circ p ^ {*} = k, $$

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 mapping $ p : X \rightarrow Y $, the greatest common divisor of all the integers $ k $ associated to a transfer is the Brouwer degree of $ p $ when $ X $ and $ Y $ are manifolds of the same dimension (cf. also Degree of a mapping). Some of these transfers are induced by $ S $- mappings, 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. 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.

References

[a1]	J.C. Becker, D.H. Gottlieb, "Vector fields and transfers" Manuscr. Math. , 72 (1991) pp. 111–130 Zbl 0736.55012