TITLE: When are homology groups the trivial group?
QUESTION [5 upvotes]: I've noticed that all the spaces $X$ whose (singular) homology I've computed or seen computed have $H_n(X)=0$ whenever $n$ is greater than the dimension of $X$. So I have the following conjecture:

Conjecture. Suppose $n$ is the least integer such that a space $X$ may be embedded into $\mathbb{R}^n$. Then for all $m >n$ we have $H_m(X)=0$. 

Is this true?

REPLY [8 votes]: This is false, and somewhat surprising. There is a counterexample due to Milnor and Barrat . The space is a union of a nested sequence of spheres embedded in $\mathbb{R}^3$ and it has infinitely many non-zero singular homology groups. The nesting of the spheres is similar as in the Hawaiian earring.
However, the result is true if one imposes local niceness on the space $X$, e.g. if $X$ is a subcomplex of an $n$-dimensional CW-complex.