We provide a sufficient condition for the nontriviality of the Lipschitz homotopy group of the Heisenberg group, $${\pi_m^{\rm Lip}(\mathbb{H}_n)}$$ π m Lip ( H n ) , in terms of properties of the classical homotopy group of the sphere, $${\pi_m(\mathbb{S}^n)}$$ π m ( S n ) . As an application we provide a new simplified proof of the fact that $${\pi_n^{\rm Lip}(\mathbb{H}_n)\neq \{0\}, n=1,2,\ldots}$$ π n Lip ( H n ) ≠ { 0 } , n = 1 , 2 , … , and we prove a new result that $${\pi_{4n-1}^{\rm Lip}(\mathbb{H}_{2n})\neq \{0\}}$$ π 4 n - 1 Lip ( H 2 n ) ≠ { 0 } for n = 1,2,… The last result is based on a new generalization of the Hopf invariant. We also prove that Lipschitz mappings are not dense in the Sobolev space $${W^{1,p}(\mathcal{M},\mathbb{H}_{2n})}$$ W 1 , p ( M , H 2 n ) when $${\dim \mathcal{M} \geq 4n}$$ dim M ≥ 4 n and 4n−1 ≤ p < 4n.