We study large groups of birational transformations $\operatorname{Bir}(X)$, where $X$ is a variety of dimension at least $3$, defined over $\mathbf{C}$ or a subfield of $\mathbf{C}$. Two prominent cases are when $X$ is the projective space $\mathbb{P}^n$, in which case $\operatorname{Bir}(X)$ is the Cremona group of rank $n$, or when $X \subset \mathbb{P}^{n+1}$ is a smooth cubic hypersurface. In both cases, and more generally when $X$ is birational to a conic bundle, we produce infinitely many distinct group homomorphisms from $\operatorname{Bir}(X)$ to $\mathbf{Z}/2$, showing in particular that the group $\operatorname{Bir}(X)$ is not perfect, and thus not simple. As a consequence, we also obtain that the Cremona group of rank $n \geqslant 3$ is not generated by linear and Jonquières elements.

For each prime $p$, we define a $t$‑structure on the category $\,\widehat{\!S^{0,0}}/\tau\text{-}\mathbf{Mod}_{\mathrm{harm}}^b$ of harmonic $\mathbb{C}$-motivic left-module spectra over $\,\widehat{\!S^{0,0}}/\tau$, whose MGL-homology has bounded Chow–Novikov degree, such that its heart is equivalent to the abelian category of $p$‑completed $\mathrm{BP}_*\mathrm{BP}$-comodules that are concentrated in even degrees. We prove that $\,\widehat{\!S^{0,0}}/\tau\text{-} \mathbf{Mod}_{\mathrm{harm}}^b$ is equivalent to $\mathcal{D}^b({\mathrm{BP}_*\mathrm{BP}\text{-}\mathbf{Comod}}^{\mathrm{ev}})$ as stable $\infty$-categories equipped with $t$‑structures.

As an application, for each prime $p$, we prove that the motivic Adams spectral sequence for $\,\widehat{\!S^{0,0}}/\tau$, which converges to the motivic homotopy groups of $\,\widehat{\!S^{0,0}}/\tau$, is isomorphic to the algebraic Novikov spectral sequence, which converges to the classical Adams–Novikov $E_2$-page for the sphere spectrum $\,\widehat{\!S^0}$. This isomorphism of spectral sequences allows Isaksen and the second and third authors to compute the stable homotopy groups of spheres at least to the $90$-stem, with ongoing computations into even higher dimensions.