We search for the JPC = 0−− and 1þ− light tetraquark states with masses up to 2.46 GeV=c2 in ϒð1SÞ and ϒð2SÞ decays with data samples of ð102 2Þ million and ð158 4Þ million events, respectively, collected with the Belle detector. No significant signals are observed in any of the studied production modes, and 90% credibility level (C.L.) upper limits on their branching fractions in ϒð1SÞ and ϒð2SÞ decays are obtained. The inclusive branching fractions of the ϒð1SÞ and ϒð2SÞ decays into final states with f1ð1285Þ are measured to be Bðϒð1SÞ → f1ð1285Þ þ anythingÞ ¼ ð46 28ðstatÞ 13ðsystÞÞ × 10−4 and Bðϒð2SÞ → f1ð1285Þ þ anythingÞ ¼ ð22 15ðstatÞ 6.3ðsystÞÞ × 10−4. The measured χb2 → J=ψ þ anything branching fraction is measured to be ð1.50 0.34ðstatÞ 0.22ðsystÞÞ × 10−3, and 90% C.L. upper limits for the χb0;b1 → J=ψ þ anything branching fractions are found to be 2.3 × 10−3 and 1.1 × 10−3, respectively. For Bðχb1 → ω þ anythingÞ, the branching fraction is measured to be ð4.9 1.3ðstatÞ 0.6ðsystÞÞ × 10−2. All results reported here are the first measurements for these modes.