Weil's group chunk theorem: A topological setting (Q1812783)

The author proves a topological version of Weil's theorem: A birational group law that is only partially defined can be extended to an algebraic group. A group chunk is a topological space with a multiplication \(\rho\) and an inversion map \(i\) defined on subsets \(U\subset X\times X\), \(V\subset X\), respectively, such that for each \(x\in X\), the left multiplication \(\lambda_ x\), defined by \(\lambda_ x=\rho(x,\cdot)\), the right multiplication, defined by \(\rho_ x=\rho(\cdot,x)\) for \(x\in V\), and the inversion \(i\) are homeomorphisms. For \(x\in X\) the set \(\{z\in X: (xz)z^{-1}\) is defined\} is a non-empty open set in \(X\), and for \(x,y,z\in X\) the identities \((xy)z=x(yz)\), \((xz)z^{-1}=x\), \(z^{- 1}(zx)=x\) hold whenever both sides are defined. The main theorem of the paper is as follows: Associated to a group chunk \((x,\rho,i)\) there is a realisation \((G,h)\), where \(G\) is a homogeneous group (a group with a topology in which inversion and translations are continuous) and \(h\) a homeomorphism from \(X\) onto a dense subset \(h[X]\) of \(G\) such that \(h(xy)=h(x)h(y)\) for all \((x,y)\) in the domain of \(\rho\). If \((G^*,h^*)\) is another realisation, then there is a unique homomorphism \(\alpha: G\to G^*\) of the abstract groups such that \(h^*=\alpha h\) and this map is then a homeomorphism between \(G\) and \(G^*\). This is a generalisation of Weil's theorem for the topological setting covering the cases of quasi-algebraic group chunks and differentially algebraic group chunks. With the help of the above theorem the author gives a positive solution to a problem in model theory, especially the characteristic \(p\) case: Whether all groups that are first-order definable in algebraically closed fields are isomorphic to algebraic groups.