A celebrated theorem of M. F. Atiyah and F. Hirzebruch [“Spin-manifolds and group actions”, Essays Topol. Relat. Top., Mem. dédiés a Georges de Rham, 18–28 (1970; Zbl 0193.52401)] states that \(\widehat{A}(M) = 0\) if \(M\) admits a smooth non-trivial \(S^1\)-action, where \(M\) denotes a smooth closed connected spin-manifold. In this paper the author considers the case where \(M\) is highly connected. Using the theory of elliptic genera, the author in fact shows that, in addition to the above one, there exist further obstructions according to the degree of connectivity of \(M\). Let \(\Phi(M)\) be a modular function of weight 0 with \(\mathbb Z_2\)-character for a subgroup \(\Gamma_0(2)\) of \(SL_2(\mathbb Z)\) consisting of all matrices whose \((2, 1)\)-components are even. Letting \(\Phi_0(M)\) denote its \(q\)-expansion \[ \Phi_0(M)=q^{-\dim M/8}\cdot (\widehat{A}(M)-\widehat{A}(M, TM)\cdot q +\widehat{A}(M, \Lambda^2TM+TM)\cdot q^2 + \cdots ) \] in the \(\widehat{A}\)-cusp, where \(\widehat{A}(M, E)\) denotes the index of the Dirac operator twisted with \(E\otimes \mathbb C\), the author proves the following theorem: Let \(M\) be be a \(k\)-connected manifold for \(k \geq 4r > 0\). If \(M\) admits a smooth non-trivial \(S^1\)-action, then the first \((r+1)\) coefficients of \(\Phi_0(M)\) vanish. This is established by considering the \(q\)-expansion \(\text{sign}_{S^1}(q, \mathcal{L}M)\) of the \(S^1\)-equivariant elliptic genus in the signature cusp, where \(\mathcal{L}M\) denotes the free loop space of \(M\). Briefly sketching, it can be stated as follows. Using the Lefschetz fixed point formula it follows from the assumption of the theorem that the first \((r+1)\) coefficients of \(\text{sign}_{S^1}(q, \mathcal{L}M)\) vanish at the involusion \(\sigma\in S^1\) as characters of \(S^1\). In addition, via the rigidity theorem one knows that \(\text{sign}_{S^1}(q, \mathcal{L}M)(\sigma)\) is equal to the \(q\)-expansion of \(\Phi(M)\) in the singular cusp, so that changing cusps one obtains the theorem. Finally the author provides an example of an 8-connected manifold with vanishing Witten genus but \(\widehat{A}(M, \Lambda^2TM+TM)\neq 0\). This example shows that the author’s theorem is independent of the vanishing theorem for the Witten genus [see the author, “\(\text{Spin}^c\)-manifolds with Pin(2)-action”, Math. Ann. 315, 511–528 (1999; Zbl 0963.19002) and K. Liu, “On modular invariance and rigidity theorems”, J. Differ. Geom. 41, 343–396 (1995; Zbl 0836.57024)].