  We present a novel technique called "photometric IGM tomography" to map the
intergalactic medium (IGM) at $z\simeq4.9$ in the COSMOS field. It utilizes
deep narrow-band (NB) imaging to photometrically detect faint Ly$\alpha$ forest
transmission in background galaxies across the Subaru/Hyper-Suprime Cam (HSC)'s
$1.8\rm\,sq.\,deg$ field of view and locate Ly$\alpha$ emitters (LAEs) in the
same cosmic volume. Using ultra-deep HSC images and Bayesian spectral energy
distribution fitting, we measure the Ly$\alpha$ forest transmission at
$z\simeq4.9$ along a large number ($140$) of background galaxies selected from
the DEIMOS10k spectroscopic catalogue at $4.98<z<5.89$ and the SILVERRUSH LAEs
at $z\simeq5.7$. We photometrically measure the mean Ly$\alpha$ forest
transmission and achieve a result consistent with previous measurements based
on quasar spectra. We also measure the angular LAE-Ly$\alpha$ forest
cross-correlation and Ly$\alpha$ forest auto-correlation functions and place an
observational constraint on the large-scale fluctuations of the IGM around LAEs
at $z\simeq4.9$. Finally, we present the reconstructed 2D tomographic map of
the IGM, co-spatial with the large-scale structure of LAEs, at a transverse
resolution of $11 \,h^{-1}\rm cMpc$ across $140\,h^{-1}\rm cMpc$ in the COSMOS
field at $z\simeq4.9$. We discuss the observational requirements and the
potential applications of this new technique for understanding the sources of
reionization, quasar radiative history, and galaxy-IGM correlations across
$z\sim3-6$. Our results represent the first proof-of-concept of photometric IGM
tomography, offering a new route to examining early galaxy evolution in the
context of the large-scale cosmic web from the epoch of reionization to cosmic
noon.
