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In this paper we use the notion of operator-valued symbol in order to computethe index of Toeplitz operators on compact Lie groups. Our approach combinesthe Connes index theorem and the infinite-dimensional operator-valued symboliccalculus of Ruzhansky-Turunen. We also give applications to the index ofToeplitz operators on simply connected closed $3$-manifolds $\mathbb{M}\simeq\mathbb{S}^3\simeq \textnormal{SU}(2) ,$ by using, as a fundamental tool, thePoincar\'e theorem (see Perelman [31,32,33,34])

The index of Toeplitz operators on compact Lie groups and simply connected closed 3-manifolds