Here is a maybe false proof that I came up with that homology of topological spaces is homotopy invariant. I'm thinking that it is indeed fake because why hasn't anyone else come up with this much simpler proof until now if it was correct?
$\textbf{Proof}:$
Let $\text{Sing}(-)$ denote the singular set functor $\textbf{Top} \rightarrow s\textbf{Set}$ then if two maps between topological spaces $f,g:X \rightarrow Y$ are homotopic by $H:X \times \Delta^1_{Top} \rightarrow Y$ where $\Delta^1_{Top}$ is the 1-simplex as a topological space.
Then $f_{\#},g_{\#}:\text{Sing}(X) \rightarrow \text{Sing}(Y)$ are homotopic by the simplicial homotopy $$\text{Sing}(X) \times \Delta^1 \xrightarrow{id \times \phi} \text{Sing}(X) \times \text{Sing}(\Delta^1_{Top}) \xrightarrow{H_{\#}} \text{Sing}(Y)$$ where $\phi: \Delta^1 \rightarrow \text{Sing}(\Delta^1_{Top})$ is the obvious map and $\text{Sing}(X) \times \text{Sing}(\Delta^1_{Top}) = \text{Sing}(X \times \Delta^1_{Top})$.
And since simplicial homology is homotopy invariant $f_*=g_*:H_*(X) \rightarrow H_*(Y)$
$\textbf{Finished}$
The classic proof of homotopy invariance of the homology functor relies on the acyclicity of the $\Delta^n_{Top}$ w.r.t the homology functor, something which isn't used here. That's what leads me to believe that this is indeed false. I haven't been able to spot the error yet though.
Also, the map $\phi$ is explicitly the unit of the adjunction $\mid -\mid \dashv \text{Sing}(-)$, that is if $\eta: \text{id} \rightarrow \text{Sing}(\mid - \mid)$ then $\eta_{\Delta^1} = \phi$
Thanks for helping me out in advance!