John Lott has computed an integer-valued signature for the orbit space of a compact orientable $$(4k+1)$$ manifold with a semi-free $$S^1$$-action, which is a homotopy invariant of that space, but he did not construct a Dirac type operator which has this signature as its index. In this Thesis, we construct such operator on the orbit space, a Thom-Mather stratified space with one singular stratum of positive dimension, and we show that it is essentially unique and that its index coincides with Lott’s signature, at least when the stratified space satisfies the so called Witt condition. We call this operator the induced Dirac-Schrödinger operator. The strategy of the construction is to “push down” an appropriate $$S^1$$-invariant first order transversally elliptic operator to the quotient space.