Can a Layman Expose the Eucharist?

In certain circumstances, yes. The Code of Canon Law:

The minister of exposition of the Blessed Sacrament and of the eucharistic blessing is a priest or deacon. In special circumstances, the minister of exposition and deposition alone, but without the blessing, is an acolyte, and extraordinary minister of holy Communion, or another person deputed by the local ordinary, in accordance with the regulations of the diocesan bishop (can. 943).

In other words, exposition of the Blessed Sacrament is to be performed by a priest or deacon. When this isn’t possible, a layperson may expose and repose the Blessed Sacrament, but there should not be any blessing.