It is a standard consequence of Hurewicz's theorem that a homology eqivalence between simply connected spaces is a weak equivalence (and hence a homotopy equivalence, if the spaces are CW-complexes).
What is more, it is even enough to assume that the map is a homology equivalence with local coefficients and an iso on $\pi_1$, see e.g. this question.
(On the other hand, a map which is only a homology equivalence does not need to be an isomorphism on $\pi_1$ and hence not a weak equivalence.)
What is a concrete example of a map that is an isomorphism on homology with $\mathbb Z$ coefficients, and also an isomorphism on $\pi_1$, but not a weak equivalence?
Best Answer
Let $X$ be the CW complex obtained from $S^1 \vee S^n$, $n>1$, by attaching an $(n+1)$-cell via a map $S^n\to S^1\vee S^n$ representing the element $2t-1$ in $\pi_n(S^1\vee S^n) \cong {\mathbb Z}[t,t^{-1}]$, so $\pi_n(X)\cong{\mathbb Z}[t,t^{-1}]/(2t-1)\cong {\mathbb Z}[1/2]\subset{\mathbb Q}$. The inclusion map $S^1 \to X$ then induces an isomorphism on homology and on $\pi_i$ for $i<n$ but not on $\pi_n$.
This example relies heavily on the nontriviality of the action of $\pi_1$ on $\pi_n$, but this is necessary since Whitehead's theorem that homology isomorphisms between simply-connected CW complexes are homotopy equivalences holds more generally for CW complexes with trivial action of $\pi_1$ on all homotopy groups including $\pi_1$. (This is Proposition 4.74 in my Algebraic Topology book, and the example is Example 4.35. Sorry for the self-references, but I should know the book as well as anyone!)