ring the fitting procedure (normal 1/error$^{2}$ weighting).\
[*Choice of emissivity -* ]{}In order to derive an estimate of the emissivity index value of the complex, we first let the emissivity index of the cold dust component vary ($\beta$$_c$ free) and perform the two-MBB fitting for each ISM resolved elements. We obtain an average emissivity index of 1.47 $\pm$ 0.17 across the complex, consistent with the results derived for the whole LMC by @Planck_collabo_2011_MagellanicClouds ($\sim$1.5) or for a strip in the LMC using [[*Herschel*]{}]{} Science Demonstration Phase data by @Gordon2010. We thus decide to fix the effective emissivity index to 1.5 and re-derive the temperature map. Fixing the emissivity index value prevents biases linked with [*1)*]{} temperature-emissivity anti-correlation due to measurement uncertainties, [*2)*]{} temperature mixing along the line of sight contributing to a bias in the derived “effective emissivity index" [see @Juvela2012; @Galametz2012 for discussions on these issues].\
[*Error estimates -* ]{}We perform Monte Carlo simulations in order to quantify the uncertainties on dust temperatures driven by flux errors of the resolved fluxes. Error maps are a combination in quadrature of the uncertainty maps derived during the data reduction (mapping, pointing) and uncertainties in the absolute calibration (15$\%$ for the SPIRE bands that are the most crucial for dust mass estimates), the latter being the dominant source of uncertainties. We generate 20 sets of modified constraints with fluxes randomly varying within their error bars and following a normal distribution around the nominal value and run the model for each set. Resolved uncertainties can thus derived using the standard deviations of each distribution. Since the absolute calibration is correlated for SPIRE bands, the three SPIRE measurements move consistently in each set.\

Realistic dust properties fitting
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We also apply the phenomenological SED fitting procedure of @Galliano2011 to