We establish inequalities that constrain the genera of smooth cobordisms between knots in 4-dimensional cobordisms. These “relative adjunction inequalities” improve for closed surfaces which have been instrumental many topological applications gauge theory. The relative refine latter by incorporating numerical invariants boundary associated to Heegaard Floer homology classes determined 4-manifo...

In this paper, we prove an $\mathbb{A}^1$-homology version of the Whitehead theorem with dimension bound. We also excision for $\mathbb{A}^1$-homology, Suslin homology, and $\mathbb{A}^1$-homotopy sheaves. order to these results, develop a general theory relative sheaves establish weak comparison result between $\mathbb{A}^1$- homology. As example as application results above, compute hyperplan...