Equal homology does not imply homotopy equivalence

By [A1], the spaces $X$ and $Y$ have isomorphic singular homology groups in every degree, so the refuted claim's hypothesis holds for this pair.

Write $W_2:=S^1\vee S^1$ and let $q:W_2\bigsqcup S^2\to Y$ be the quotient map from [L7], with both basepoints identified to $y_0\in Y$. Choose a small open disk $D\subseteq S^2$ around the sphere basepoint and small open arcs $O_1,O_2$ around the wedge point in the two circle summands of $W_2$. Put $U:=q(W_2\bigsqcup D),V:=q((O_1\cup O_2)\bigsqcup S^2)\mathrm{.}$ By the quotient description in [L7], these are open path-connected subsets of $Y$ covering $Y$. The set $U$ deformation retracts onto $W_2$ by contracting $D$ to $y_0$, the set $V$ deformation retracts onto $S^2$ by contracting the two arcs to $y_0$, and the overlap $U\cap V=q((O_1\cup O_2)\bigsqcup D)$ deformation retracts onto $y_0$.

By [L1], ${\pi }_1(X,([0],[0]))\cong {\mathbb{Z}}^2$. The group ${\mathbb{Z}}^2$ is abelian, while $F_2$ is not abelian because the reduced words $ab$ and $ba$ are distinct. Hence ${\pi }_1(X,([0],[0]))\cong ̸{\pi }_1(Y,y_0)$.

By [L2] and [L5], the deformation retraction in step 1.2 gives ${\pi }_1(U,y_0)\cong F_2$. By [L3] and [L5], the corresponding deformation retraction gives ${\pi }_1(V,y_0)=1$. Since $U\cap V$ deformation retracts to the point $y_0$, it is simply connected. Applying [L4] to the open cover $Y=U\cup V$ therefore yields ${\pi }_1(Y,y_0)\cong {\pi }_1(U,y_0)\ast {\pi }_1(V,y_0)\cong F_2\mathrm{.}$

If $X$ and $Y$ were homotopy equivalent, choose homotopy inverse maps $h:X\to Y$ and $k:Y\to X$, together with homotopies $H:k\circ h\simeq {\mathrm{id}}_X$ and $K:h\circ k\simeq {\mathrm{id}}_Y$. Put $\beta (t):=H(([0],[0]),t)$ and $\gamma (t):=K(y_0,t)$. [L6, step 1.3, construct] For any path $\rho$ from $a$ to $b$, the assignment $[\alpha ]\mapsto [\bar{\rho }\ast \alpha \ast \rho ]$ transports loop classes from ${\pi }_1(X,a)$ to ${\pi }_1(X,b)$, and reversing $\rho$ gives the inverse transport because the inserted pairs $\rho \ast \bar{\rho }$ and $\bar{\rho }\ast \rho$ contract to constant loops. Applying this to $\beta$ and $\gamma$ yields isomorphisms ${\beta }_{\mathrm{\#}}:{\pi }_1(X,k(h(([0],[0]))))\stackrel{\sim }{\to }{\pi }_1(X,([0],[0])),{\gamma }_{\mathrm{\#}}:{\pi }_1(Y,h(k(y_0)))\stackrel{\sim }{\to }{\pi }_1(Y,y_0)\mathrm{.}$ The maps $h:(X,([0],[0]))\to (Y,h(([0],[0])))$ and $k:(Y,y_0)\to (X,k(y_0))$ are pointed, so [L6] gives induced homomorphisms on fundamental groups. Precomposing $H$ and $K$ with based loops and then transporting the moving basepoints by ${\beta }_{\mathrm{\#}}$ and ${\gamma }_{\mathrm{\#}}$ shows that these induced homomorphisms are inverse isomorphisms up to the two basepoint transports. Therefore ${\pi }_1(X,([0],[0]))\cong {\pi }_1(Y,y_0)$, contradicting step 1.3.

Therefore $X$ and $Y$ are not homotopy equivalent even though step 1.1 shows that their singular homology groups agree.