The induced character formula in classical representation theory can be used, among other things, to describe the dimension of coinvariants of a representation in terms of its character. In this talk, I will explain how this formula is related to the multiplicativity of Euler characteristics in algebraic topology, and, in a more homotopy-coherent context, to the composability of so-called Becker-Gottlieb transfers, which are "wrong-way maps" in singular (co)homology; by describing a general formula to compute "dimensions of homotopy colimits". If time permits, I will discuss the most general case in which composability of Becker-Gottlieb transfers is now known. This is based on joint works with Carmeli-Cnossen-Yanovski, Klein-Malkiewich (and the last part with Volpe-Wolf).