#### Volume 20, issue 5 (2020)

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

### João Paulo Santos

Algebraic & Geometric Topology 20 (2020) 2177–2268
##### Abstract

We study the moduli space ${\mathfrak{𝔐}}_{k}^{r}\left({\stackrel{̃}{ℙ}}_{q}^{2}\right)$ of rank $r$ holomorphic bundles with trivial determinant and second Chern class ${c}_{2}=k$, over the blowup ${\stackrel{̃}{ℙ}}_{q}^{2}$ 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 ${\mathfrak{𝔐}}_{k}^{r}\left({\stackrel{̃}{ℙ}}_{q}^{2}\right)$ and the degree $k$ component of the bar construction $B\left({\mathfrak{𝔐}}^{r}{ℙ}^{2},{\left({\mathfrak{𝔐}}^{r}{ℙ}^{2}\right)}^{q},{\left({\mathfrak{𝔐}}^{r}{\stackrel{̃}{ℙ}}_{1}^{2}\right)}^{q}\right)$. The space ${\mathfrak{𝔐}}_{k}^{r}\left({\stackrel{̃}{ℙ}}_{q}^{2}\right)$ is isomorphic to the moduli space $\mathfrak{𝔐}{\mathsc{ℐ}}_{k}^{r}\left({X}_{q}\right)$ of charge $k$ based $SU\left(r\right)$ instantons on a connected sum ${X}_{q}$ of $q$ copies of $\overline{{ℙ}^{2}}$ and we show that, for $k=1,2$, we have a homotopy equivalence between $\mathfrak{𝔐}{\mathsc{ℐ}}_{k}^{r}\left({X}_{q}#{X}_{s}\right)$ and the degree $k$ component of $B\left(\mathfrak{𝔐}{\mathsc{ℐ}}^{r}\left({X}_{q}\right),\mathfrak{𝔐}{\mathsc{ℐ}}^{r}\left({S}^{4}\right),\mathfrak{𝔐}{\mathsc{ℐ}}^{r}\left({X}_{s}\right)\right)$. 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.

##### Keywords
moduli space, holomorphic bundles, instantons, bar construction
##### Mathematical Subject Classification 2010
Primary: 14D21, 58D27
Secondary: 14J60, 55P48