Xin Peng

Maximal first Betti number drop and collapsing RCD spaces

Let $(X_i,d_i,\mathfrak m_i)$ be compact $\mathrm{RCD}(K,N)$ spaces converging to a space $X$ of rectifiable dimension $m$. Their first Betti numbers can drop by at most $N-m$. When equality holds, the maximal abelian covers have a non-collapsed limit $Y$ carrying a free isometric $\mathbb{R}^{N-m}$-action. We turn thi...