Context. Superclusters of galaxies and their surrounding low-density regions (cocoons) represent dynamically evolving environments in which galaxies and their systems form and evolve. While evolutionary processes of galaxies in dense environments are extensively studied at present, galaxy evolution in low-density regions has received less attention. Aims. We study the properties, connectivity, and galaxy content of groups and filaments in the A2142 supercluster (SCl A2142) cocoon to understand the evolution of the supercluster with its surrounding structures and the galaxies within them. Methods. We calculated the luminosity-density field of SDSS galaxies and traced the SCl A2142 cocoon boundaries by the lowest luminosity-density regions that separate SCl A2142 from other superclusters. We determined galaxy filaments and groups in the cocoon and analysed the connectivity of groups, the high density core (HDC) of the supercluster, and the whole of the supercluster. We compared the distribution and properties of galaxies with different star-formation properties in the supercluster and in the cocoon. Results. The supercluster A2142 and the long filament that is connected to it forms the longest straight structure in the Universe detected so far, with a length of approximately 75 h −1 Mpc. The connectivity of the cluster A2142 and the whole supercluster is C = 6 − 7; poor groups exhibit C = 1 − 2. Long filaments around the supercluster's main body are detached from it at the turnaround region. Among various local and global environmental trends with regard to the properties of galaxies and groups, we find that galaxies with very old stellar populations lie in systems across a wide range of richness from the richest cluster to poorest groups and single galaxies. They lie even at local densities as low as D1 < 1 in the cocoon and up to D1 > 800 in the supercluster. Recently quenched galaxies lie in the cocoon mainly in one region and their properties are different in the cocoon and in the supercluster. The star-formation properties of single galaxies are similar across all environments. Conclusions. The collapsing main body of SCl A2142 with the detached long filaments near it are evidence of an important epoch in the supercluster evolution. There is a need for further studies to explore possible reasons behind the similarities between galaxies with very old stellar populations in extremely different environments, as well as mechanisms for galaxy quenching at very low densities. The presence of long, straight structures in the cosmic web may serve as a test for cosmological models.