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](https://ncatlab.org/nlab/show/Becker-Gottlieb+transfer). Here are a few of them:

* <cite authors="Becker, James C.; Gottlieb, Daniel H.">_Becker, James C.; Gottlieb, Daniel H._, [**The transfer map and fiber bundles**](http://dx.doi.org/10.1016/0040-9383(75)90029-4), Topology 14, 1-12 (1975). [ZBL0306.55017](https://zbmath.org/?q=an:0306.55017).</cite>
* <cite authors="Becker, James C.; Gottlieb, Daniel Henry">_Becker, James C.; Gottlieb, Daniel Henry_, [**Vector fields and transfers**](http://dx.doi.org/10.1007/BF02568269), Manuscr. Math. 72, No.2, 111-130 (1991). [ZBL0736.55012](https://zbmath.org/?q=an:0736.55012).</cite>
* *Klein, John; Malkiewich, Cary*, [**The transfer is functorial**](https://arxiv.org/abs/1603.01872), arXiv:1603.01872.