We examine the HI-to-stellar mass ratio (HI fraction) for galaxies near filament backbones within the nearby Universe ($d <$ 181 Mpc). This work uses the 6 degree Field Galaxy Survey and the Discrete Persistent Structures Extractor to define the filamentary structure of the local cosmic web. HI spectral stacking of HI Parkes all sky survey observations yields the HI fraction for filament galaxies and a field control sample. The HI fraction is measured for different stellar masses and fifth nearest neighbour projected densities ($Sigma_{5}$) to disentangle what influences cold gas in galaxies. For galaxies with stellar masses log($M_{star}$) $<$ 11 M$_{odot}$ in projected densities 0 $leq$ $Sigma_{5}$ $<$ 3 galaxies Mpc$^{-2}$, all HI fractions of galaxies near filaments are statistically indistinguishable from the control sample. Galaxies with stellar masses log($M_{star}$) $geq$ 11 M$_{odot}$ have a systematically higher HI fraction near filaments than the control sample. The greatest difference is 0.75 dex, which is 5.5$sigma$ difference at mean projected densities of 1.45 galaxies Mpc$^{-2}$. We suggest that this is evidence for massive galaxies accreting cold gas from the intrafilament medium that can replenish some HI gas. This supports cold mode accretion where filament galaxies with a large gravitational potential can draw gas from the large-scale structure.
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