Context. A large fraction of the interstellar medium can be characterized as a multiphase medium. The neutral hydrogen gas is bistable with a cold and warm neutral medium (CNM and WNM respectively) but there is evidence for an additional phase at intermediate temperatures, a lukewarm neutral medium (LNM) that is thermally unstable. Aims. We use all sky data from the HI4PI survey to separate these neutral HI phases with the aim to determine their distribution and phase fractions in the local interstellar medium. Methods. HI4PI observations, gridded on an nside = 1024 HEALPix grid, were decomposed into Gaussian components. From the frequency distribution of the velocity dispersions we infer three separate linewidth regimes. Accordingly we extract the HI line emission corresponding to the CNM, LNM, and WNM. We generated all-sky maps of these phases in the local HI gas with -8 < v_LSR < 8 km/s. Results. Each of the HI phases shows distinct structures on all scales. The LNM never exists as a single phase but contributes on average 41% of the HI. The CNM is prominent only for 22% of the sky, contributes there on average 34% but locally up to 60% of the HI and is associated with dust at temperatures T_dust ~ 18.6 K. Embedded cold filaments show a clear anti-correlation between CNM and LNM. Also the smoothly distributed WNM is anti-correlated with the CNM. It contributes for the rest of the sky 39% with dust associated at temperatures T_dust ~ 19.4 K. Conclusions. The CNM in filaments exists on small scales. Here the observed anti-correlation between LNM and CNM implies that both, filaments and the surrounding more extended LNM, must have a common origin.
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