  We study the lift-and-project procedures of Lov{\'a}sz-Schrijver and
Sherali-Adams applied to the standard linear programming relaxation of the
traveling salesperson problem with triangle inequality. For the asymmetric TSP
tour problem, Charikar, Goemans, and Karloff (FOCS 2004) proved that the
integrality gap of the standard relaxation is at least 2. We prove that after
one round of the Lov{\'a}sz-Schrijver or Sherali-Adams procedures, the
integrality gap of the asymmetric TSP tour problem is at least 3/2, with a
small caveat on which version of the standard relaxation is used. For the
symmetric TSP tour problem, the integrality gap of the standard relaxation is
known to be at least 4/3, and Cheung (SIOPT 2005) proved that it remains at
least 4/3 after $o(n)$ rounds of the Lov{\'a}sz-Schrijver procedure, where $n$
is the number of nodes. For the symmetric TSP path problem, the integrality gap
of the standard relaxation is known to be at least 3/2, and we prove that it
remains at least 3/2 after $o(n)$ rounds of the Lov{\'a}sz-Schrijver procedure,
by a simple reduction to Cheung's result.
