© 1996, Topology Atlas

Many kinds of homotopy, homology and cohomology groups are defined for arbitrary spaces. However, there have been little investigation about calculation of such groups for wild spaces, i.e. spaces which are far from manifolds. One reason is that algebra has been developed mainly for finitely generated objects and the other is that there are still many important questions in topology which are related to finitely generated groups.

The interest of calculation of such groups for wild spaces firstly the desire to determine homotopy, homology and cohomology groups for well-known wild spaces like the Hawaiian earring, Sierpinski gasket, Menger sponge and so on. Secondly, the study of such groups would give us a new and right direction for investigation of infinite groups. Infinite groups, commutative or non-commutative, have been studied mostly in their own regions and not so many applications are known. If we start from investigation of such groups, we possibly find new group theoretic phenomenons and have applications at hand. Since General Topology and Set Theory are good at treating with infinity, the connection of Algebra and these areas would give us a new developement of Algebra.

As a good example, we remind of the Specker phenomenon. (The name Specker phenomenon itself is new and due to A. Blass [1].) The Specker phenomenon was discovered by E. Specker [13] during his study of cohomology groups. The Specker theorem says that the homomorphism group from the direct product of countably-many copies of the integer group $Z$ to $Z$, i.e. $Hom(Z^\omega,Z)$, is the free abelian group generated by the projections. It is a surprising theorem, if we compare it to the fact that the cardinality of Hom(Z^\omega, Q) is $2^{continuum}$. The Specker theorem has been generalized to many directions:

• The non-abelin version due to G. Higman [10];
• The well-known theory of slender abelian groups, which is related to the existence of a measurable cardinal [8];
• Constructions of abelian groups of certain properties concerning $Z$-dual group [12];
• A recent relationship to set theory due to Blass [1], Brendle and Shelah.

[1] A. Blass, Cardinal characteristics and the product of countably many infinite cyclic groups, J. Algebra 169 (1994), 512--540.

[2] K. Eda, First countability and local simple connectivity of one point unions, Proc. Amer. Math. Soc. 109 (1990), 237--241.

[3] K. Eda, The first integral singular homology groups of one point unions, Quart. J. Math. Oxford 42 (1991), 443--456.

[4] K. Eda, Free $\sigma$-products and noncommutatively slender groups, J. Algebra 148 (1992), 243--263.

[5] K. Eda, A locally simply connected space and fundamental groups of one point unions of cones, Proc. Amer. Math. Soc. 116 (1992), 239--250.

[6] K. Eda, Free $\sigma$-products and fundamental groups of subspaces of the plane, preprint.

[7] K. Eda and K. Sakai, A factor of singular homology, Tsukuba J. Math. {\bf 15 (1991), 351--387.

[8] L. Fuchs, Infinite abelian groups vol. 1,2, Academic Press, 1970,1973.

[9] H. B. Griffiths, Infinite products of semigroups and local connectivity, Proc. London Math. Soc. 6 (1956), 455--485.

[10] G. Higman, Unrestricted free products and variety of topological groups, J. London Math. Soc. 27 (1952), 73--81.

[11] J. W. Morgan and I. A. Morris, A van kampen theorem for weak joins, Proc. London Math. Soc. 53 (1986), 562--576.

[12] G. Schlitt, Sheaves of abelian groups and the quotients $A^{**}/A$, J. Algebra 158 (1993), 50--60.

[13] E. Specker, Additive gruppen von folgen ganzer zahlen, Portugal. Math. 9 (1950), 131--140.