In this paper we use the fact that the rings of integer matrices have the power-substitution property in order to obtain a power-cancellation property for homotopy types of CW-complexes with one cell in dimensions 0 and 4n and a finite number of cells in dimension 2n.
We describe, for any compact connected Lie group G and any prime p, the monoid of self maps → which are rational equivalences. Here, denotes the p-adic completion of the classifying space of G. Among other things, we show that two such maps are homotopic if and only if they induce the same homomorphism in rational cohomology, if and only if their restrictions to the classifying space of the maximal torus of G are homotopic.
We obtain two classifications of weighted projective spaces: up to hoeomorphism and up to homotopy equivalence. We show that the former coincides with Al Amrani's classification up to isomorphism of algebraic varieties, and deduce the latter by proving that the Mislin genus of any weighted projective space is rigid.
For the Brown-Peterson spectrum BP at the prime 3, denotes Hazewinkel’s second polynomial generator of . Let denote the Bousfield localization functor with respect to . A typical example of type one finite spectra is the mod 3 Moore spectrum M. In this paper, we determine the homotopy groups for the 8 skeleton X of BP.