The Brouwer invariance theorems in reverse mathematics

In his book, John Stillwell wrote "finding the exact strength of the Brouwer invariance theorems seems to me one of the most interesting open problems in reverse mathematics." In this article, we solve Stillwell's problem by showing that (some forms of) the Brouwer invariance theorems are equivalent to weak K\"onig's lemma over the base system ${\sf RCA}_0$. In particular, there exists an explicit algorithm which, whenever weak K\"onig's lemma is false, constructs a topological embedding of $\mathbb{R}^4$ into $\mathbb{R}^3$.