is a carrier as well. By Corollary \[cor:open star swelling\], $\operatorname{ost}_{\beta K} \mathcal{K}$ is an $AE(n)$-cover. By the Carrier Theorem, there exists a map $\tilde \psi \colon S^m \to K$ that is carried by $D'$. Observe that if $x \in G \in {{\mathcal{G}_{}}}$, then $\tilde \psi(x) \in \operatorname{ost}_{\beta_K} K_{i_G}$ and $\psi(x) \in \operatorname{ost}_{\beta L} L_{i_G}$. Since $p(K_{i_G}) \subset L_{i_G}$, the maps $\psi$ and $p \circ \tilde \psi$ are $\operatorname{ost}_{\beta L} \mathcal{L}$-close. By Corollary \[cor:open star swelling\], $\operatorname{ost}_{\beta L} \mathcal{L}$ is an $AE(n)$-cover and by Lemma \[lem:uclose uhomotopic\], $\psi$ and $p \circ \tilde \psi$ are homotopic. Hence $p$ induces epimorphisms on homotopy groups of dimensions less than $n$. This concludes the second part of the proof.

Proof of Theorem \[thm:nregular pullback\]
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\[lem:simplicial regular\] Let $K$ and $L$ be polyhedra. If $p \colon K \to L$ is a simplicial, surjective map such that for each $\delta \in \tau(L)$ the inverse image $p^{-1}(\delta)$ is an $AE(n)$, then $p$ is a weak $n$-homotopy equivalence.

Apply Theorem \[thm:nerve theorem\] with the cover $\mathcal{L} = \tau(L)$ of $L$ and the cover $\mathcal{K} = p^{-1}(\mathcal{L})$ of $K$.

(Proof of Theorem \[thm:nregular pullback\]) The map $p$ is simplicial onto $\beta K$ and satisfies the conditions of Lemma 