In the more general version with compact (finitely dominated) fibers, this is called the Becker-Gottlieb transfer. You can find a long list of references on the nlab. Here are a few of them:
- Becker, James C.; Gottlieb, Daniel H., The transfer map and fiber bundles, Topology 14, 1-12 (1975). ZBL0306.55017.
- Becker, James C.; Gottlieb, Daniel Henry, Vector fields and transfers, Manuscr. Math. 72, No.2, 111-130 (1991). ZBL0736.55012.
- Klein, John; Malkiewich, Cary, The transfer is functorial, arXiv:1603.01872.