On the $K$-theory of pushouts

We reveal a relation between the behaviour of localizing invariants $E$ on pushouts and on pullbacks of ring spectra. More concretely, we show that the failure of $E$ sending a pushout of ring spectra to a pushout is controlled by the value of $E$ on a pullback of ring spectra. Vice versa, in many situations, we show that the failure $E$ of sending a pullback square to a pullback is controlled by the value of $E$ on a pushout of ring spectra. The latter can be interpreted as identifying the $\odot$-ring, introduced in earlier work of ours, as a pushout which turns out to be explicitly computable in many cases. This opens up new possibilities for direct computations. As further applications, we give new proofs of (generalizations) of Waldhausen's seminal results on the $K$-theory of generalized free products and obtain a general relation between the value of a localizing invariant on trivial square zero extensions and on tensor algebras.