Hello everyone
Is there any map on Living Atlas that can show the live or old road constructions
Living Atlas does include a global live traffic service which includes information on current traffic conditions and things that may affect traffic, like construction. See the World Traffic Service item description for more details.
Beyond the above, Living Atlas does not include current or completed highway/road construction projects. That information is best obtained from your local department of transportation. For example, if you were seeking information about California road construction projects, Caltrans would be the source for the best information.