Holomorphic bundles on the blown-up plane and the bar construction

We study the moduli space ${\mathfrak{M}}_k^r({\tilde{\mathbb{P}}}_{\text{\hspace{0.17em}⁣}q}^2)$ of rank $r$ holomorphic bundles with trivial determinant and second Chern class $c_2=k$, over the blowup ${\tilde{\mathbb{P}}}_{\text{\hspace{0.17em}⁣}q}^2$ of the projective plane at $q$ points, trivialized on a rational curve. We show that, for $k=1,2$, we have a homotopy equivalence between ${\mathfrak{M}}_k^r({\tilde{\mathbb{P}}}_{\text{\hspace{0.17em}⁣}q}^2)$ and the degree $k$ component of the bar construction $$\mathrm{B}$\bigl(\mathfrak M^r\mathbb P^2,(\mathfrak M^r\mathbb P^2)^{q},(\mathfrak M^r\tilde{\mathbb P}_{!1}^2)^{q}\bigr)$. The space ${\mathfrak{M}}_k^r({\tilde{\mathbb{P}}}_{\text{\hspace{0.17em}⁣}q}^2)$ is isomorphic to the moduli space $\mathfrak{M}{\mathcal{I}}_k^r(X_q)$ of charge $k$ based $SU(r)$ instantons on a connected sum $X_q$ of $q$ copies of $\overline{{\mathbb{P}}^2}$ and we show that, for $k=1,2$, we have a homotopy equivalence between $\mathfrak{M}{\mathcal{I}}_k^r(X_q\mathrm{\#}{X}_s)$ and the degree $k$ component of $$\mathrm{B}$\bigl(\mathfrak M\mathcal I^r(X_q),\mathfrak M\mathcal I^r(S^4),\mathfrak M\mathcal I^r(X_s)\bigr)$. Analogous results hold in the limit when $k\to \mathrm{\infty }$. As an application we obtain upper bounds for the cokernel of the Atiyah-Jones map in homology, in the rank-stable limit.