For ${\displaystyle n\geq 2}$, if X is (that is: ${\displaystyle \pi _{i}(X)=0}$ for all ${\displaystyle i<n}$), then ${\displaystyle {\tilde {H_{i}}}(X)=0}$ for all ${\displaystyle i<n}$, and the Hurewicz map ${\displaystyle h_{*}\colon \pi _{n}(X)\to H_{n}(X)}$ is an isomorphism.^{[1]: 366, Thm.4.32} This implies, in particular, that the homological connectivity equals the homotopical connectivity when the latter is at least 1. In addition, the Hurewicz map ${\displaystyle h_{*}\colon \pi _{n+1}(X)\to H_{n+1}(X)}$ is an epimorphism in this case.^{[1]: 390, ?}

For ${\displaystyle n=1}$, the Hurewicz homomorphism induces an ${\displaystyle {\tilde {h}}_{*}\colon \pi _{1}(X)/[\pi _{1}(X),\pi _{1}(X)]\to H_{1}(X)}$, between the abelianization of the first homotopy group (the fundamental group) and the first homology group.

Brown, Ronald (1989), "Triadic Van Kampen theorems and Hurewicz theorems", Algebraic topology (Evanston, IL, 1988), Contemporary Mathematics, vol. 96, Providence, RI: American Mathematical Society, pp. 39–57, :10.1090/conm/096/1022673, ISBN 9780821851029, MR 1022673

Brown, Ronald; Higgins, P. J. (1981), "Colimit theorems for relative homotopy groups", Journal of Pure and Applied Algebra, 22: 11–41, :10.1016/0022-4049(81)90080-3, ISSN 0022-4049

Brown, R.; Loday, J.-L. (1987), "Homotopical excision, and Hurewicz theorems, for n-cubes of spaces", Proceedings of the London Mathematical Society, Third Series, 54: 176–192,  10.1.1.168.1325, doi:10.1112/plms/s3-54.1.176, ISSN 0024-6115

Brown, R.; Loday, J.-L. (1987), "Van Kampen theorems for diagrams of spaces", , 26 (3): 311–334, doi:10.1016/0040-9383(87)90004-8, ISSN 0040-9383