This work presents the first integral field spectroscopy of the Homunculus nebula around \eta Carinae in the near-infrared spectral region ( J band ) .
We confirmed the presence of a hole on the polar region of each lobe , as indicated by previous near-IR long-slit spectra and mid-IR images .
The holes can be described as a cylinder of height ( i.e .
the thickness of the lobe ) and diameter of 6.5 and 6.0 \times 10 ^ { 16 } cm , respectively .
We also mapped the blue-shifted component of He i \lambda 10830 seen towards the NW lobe .
Contrary to previous works , we suggested that this blue-shifted component is not related to the Paddle but it is indeed in the equatorial disc .

We confirmed the claim of and showed that the spatial extent of the Little Homunculus matches remarkably well the radio continuum emission at 3 cm , indicating that the Little Homunculus can be regarded as a small H ii region .
Therefore , we used the optically-thin 1.3 mm radio flux to derive a lower limit for the number of Lyman-continuum photons of the central source in \eta Car .
In the context of a binary system , and assuming that the ionising flux comes entirely from the hot companion star , the lower limit for its spectral type and luminosity class ranges from O5.5 iii to O7 i .
Moreover , we showed that the radio peak at 1.7 arcsec NW from the central star is in the same line-of-sight of the ‘ Sr-filament ’ but they are obviously spatially separated , while the blue-shifted component of He i \lambda 10830 may be related to the radio peak and can be explained by the ultraviolet radiation from the companion star .