Possible homology of closed manifolds.

 March 9, 2008 mathematics

\newcommand{\N}{\mathbb{N}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}}

\newenvironment{question}[1][]{\par\textbf{Question (#1).}}{} \newenvironment{theorem}[1][]{\par\textbf{Theorem (#1).}}{} \newenvironment{lemma}[1][]{\par\textbf{Lemma (#1).}}{} \newenvironment{proof}{\textit{Proof.}}{}

In this fun paper [1] it is pointed out that

  • homology is a very basic invariant, and
  • closed manifolds are very basic objects

and so a very basic question is: what sequences of abelian groups are the homology groups of a closed simply connected manifold?

It isn’t very hard to realize any sequence of abelian groups up to the middle dimension, but that middle dimension is tricky (e.g., classify (n-1) -connected 2n -manifolds).

Anyway, I was wondering: is this realization question solvable for homology with coefficients in \mathbb{Z}/2\mathbb{Z} or \mathbb{Q} ?

[1] M. Kreck, An inverse to the Poincaré conjecture, Arch. Math. (Basel). 77 (2001) 98–106. https://doi.org/10.1007/PL00000470.