An Example of Anomalous Singular Homology

Introduction. The Cech homology groups of an r-dimensional space are known to vanish in dimensions greater than r. We will show that the corresponding assertion for the singular homology groups is false, even on the category of locally (r—1)-connected compacta. An rdimensional compactum X, locally contractible save at one point, is given such that the singular homology groups Hq(X; Q) with rational coefficients are nontrivial for infinitely many values of q. This settles a question raised by Eilenberg and Steenrod [l, Problem 22]. Let X denote the union of a countable number of r-spheres (r> 1) with a single point in common and a metric topology such that the diameter of the spheres tends to zero with increasing index. This example was suggested by Steenrod. Our result is