On the second homotopy module of two-dimensional CW complexes

Dyer, M. N.

doi:10.1090/S0002-9939-1976-0410739-6

On the second homotopy module of two-dimensional CW complexes
HTML articles powered by AMS MathViewer

by M. N. Dyer PDF

Proc. Amer. Math. Soc. 55 (1976), 400-404 Request permission

Abstract:

Let $X$ be a connected $2$-dimensional ${\text {CW}}$ complex. This note reproves from a very simple point of view two classical theorems of H. Hopf relating the homology of the fundamental group $\pi = {\pi _1}(X)$ of $X$ and the Hurewicz map on ${\pi _2}(X)$. This point of view also allows the dual theorems to be proved. If $\pi$ is a finite group, a new interpretation is given for ${H_i}(\pi ;Z)(i = 2,3)$ in terms of ${\pi _2}(X)$.

References

Henri Cartan and Samuel Eilenberg, Homological algebra, Princeton University Press, Princeton, N. J., 1956. MR 0077480
W. H. Cockcroft, On two-dimensional aspherical complexes, Proc. London Math. Soc. (3) 4 (1954), 375–384. MR 63042, DOI 10.1112/plms/s3-4.1.375
W. H. Cockcroft and R. G. Swan, On the homotopy type of certain two-dimensional complexes, Proc. London Math. Soc. (3) 11 (1961), 194–202. MR 126271, DOI 10.1112/plms/s3-11.1.193
Micheal N. Dyer, On the $2$-realizability of $2$-types, Trans. Amer. Math. Soc. 204 (1975), 229–243. MR 380783, DOI 10.1090/S0002-9947-1975-0380783-X

The homotopy theory of

complexes

Micheal N. Dyer and Allan J. Sieradski, Trees of homotopy types of two-dimensional $\textrm {CW}$-complexes. I, Comment. Math. Helv. 48 (1973), 31–44; corrigendum, ibid. 48 (1973), 194. MR 377905, DOI 10.1007/BF02566109
P. J. Hilton and U. Stammbach, A course in homological algebra, 2nd ed., Graduate Texts in Mathematics, vol. 4, Springer-Verlag, New York, 1997. MR 1438546, DOI 10.1007/978-1-4419-8566-8
Heinz Hopf, Über die Bettischen Gruppen, die zu einer beliebigen Gruppe gehören, Comment. Math. Helv. 17 (1945), 39–79 (German). MR 12229, DOI 10.1007/BF02566234
Heinz Hopf, Beiträge zur Homotopietheorie, Comment. Math. Helv. 17 (1945), 307–326 (German). MR 12544, DOI 10.1007/BF02566250
Sze-tsen Hu, Homotopy theory, Pure and Applied Mathematics, Vol. VIII, Academic Press, New York-London, 1959. MR 0106454
Saunders MacLane, Homology, 1st ed., Die Grundlehren der mathematischen Wissenschaften, Band 114, Springer-Verlag, Berlin-New York, 1967. MR 0349792