MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

This might be well known for algebraic topologist. So I am looking for an explicit example of a 4 dimensional manifold with fundamental group isomorphic to the rationals $\mathbb Q$.

share|cite|improve this question
What do you mean by "explicit"? You can take presentation complex for $Q$, immerse it in $R^4$ and then pull-back regular neighborhood. The result is your manifold. – Misha Jul 1 '13 at 13:37
See [this stackexchange question and answers.][1] [1]:… – Igor Rivin Jul 1 '13 at 17:12
up vote 48 down vote accepted

Since any compact manifold has the homotopy type of a finite CW-complex (see this MathOverflow question: Are non-PL manifolds CW-complexes?) and $\mathbb{Q}$ is not finitely presented, the manifold $X$ you are looking for is necessarily non-compact.

An explicit construction of a non-compact three-manifold $M$ with $\pi_1(M)=\mathbb{Q}$ can be found in the paper

B. Evans and L. Moser: Solvable Fundamental Groups of Compact 3-Manifolds, Transactions of the American Mathematical Society 168 (1972), see in particular page 209.

Now it sufficies to take $X=M \times \mathbb{R}$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.