On Z_2 and S^1 free actions on spaces of cohomology type (a,b)

• Autores: Pedro L. Q. Pergher
• Localización: Houston journal of mathematics, ISSN 0362-1588, Vol. 36, Nº 1, 2010, págs. 137-146
• Idioma: inglés
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• Resumen

  • Let S1 be the group of unitary complex numbers endowed with the complex multiplication, and Z2 the cyclic group of order two. If X is a topological space, denote by Hi(X,G) its i-th cohomology group with coefficients in G. For a natural number n, suppose X is a simply connected finite CW complex satisfying Hj (X,Z)=Z if j=0, n, 2n or 3n, and Hj (X,Z)=0 otherwise; here, Z is the group of integers. Let u1, u2 and u3 generate Hn(X,Z), H2n(X,Z) and H3n(X,Z), respectively. We say that X has type (a,b), for integers a and b, if u1u1=au2 and u1u2=bu3. In this paper, we show that Z2 can not act freely on a space of type (a,b) if a is odd and b is even, and that S1 can not act freely on a space of type (a,b) if a is different from zero. For the remaining pairs (a,b), we may have free actions, and thus it makes sense to ask for the possible cohomology rings of the corresponding orbit spaces. In this direction, we determine the possible Z2-cohomology rings of orbit spaces of free actions of Z2 on spaces of type (a,b), where a and b are even, and of free actions of S1 on spaces of type (0,b). As a consequence of these cohomological calculations, we also obtain some results of the Borsuk-Ulam type, concerning the existence of equivariant maps from the m-dimensional sphere, equipped with standard G-actions (G = Z2 or S1), into X, where X is a space of type (a,b) equipped with arbitrary G-actions