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

Abstract

We study the moduli space $\mathfrak M_k^r(\tilde{\mathbb P}^2_{\!q})$ of rank $r$ holomorphic bundles with trivial determinant and second Chern class $c_2=k$, over the blowup $\tilde{\mathbb P}^2_{\!q}$ 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 $\mathfrak M_k^r(\tilde{\mathbb P}^2_{\!q})$ 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 $\mathfrak M_k^r(\tilde{\mathbb P}^2_{\!q})$ 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 $\mathfrak M\mathcal I_k^r(X_q\# X_s)$ 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 $k\to\infty$. As an application we obtain upper bounds for the cokernel of the Atiyah-Jones map in homology, in the rank-stable limit.