Parametrized scissors congruence $K$-theory of manifolds and cobordism categories

We introduce a parametrized version of scissors congruence $K$-theory of manifolds with tangential structure, which includes a topologized version of the scissors congruence $K$-theory of oriented manifolds as a special case. We examine the relation of this $K$-theory spectrum with cut-and-paste invariants, the (parametrized) cobordism category and with (bivariant) algebraic $K$-theory of spaces. We show that the scissors congruence $K$-theory of oriented manifolds agrees on $\pi_0$ with a version of the oriented cobordism category where we allow cobordisms to have free boundaries. Lastly, we show that the spectrum level refinement of the Euler characteristic from the scissors congruence $K$-theory to $K(\mathbb{Z})$ detects on $\pi_1$ the Kervaire semicharacteristic.