Homology, Homotopy and Applications

Volume 16 (2014)

Number 2

A useful lemma on equivariant maps

Pages: 307 – 309

DOI: http://dx.doi.org/10.4310/HHA.2014.v16.n2.a17


D. Gonçalves (Departamento de Matemática, IME, University of São Paulo, Brazil)

A. Skopenkov (Faculty of Innovations and High Technology, Moscow Institute of Physics and Technology, Dolgoprudnyi, Russia; and Independent University of Moscow, Russia)


We present a short proof of the following known result. Suppose $X, Y$ are finite connected CW-complexes with free involutions, $f \colon X \to Y$ is an equivariant map, and $l$ is a non-negative integer. If $f^* \colon H^i (Y) \to H^i (X)$ is an isomorphism for each $i>l$ and is onto for $i=l$, then $f^{\sharp} \colon \pi^i_{eq}(Y)\to \pi^i_{eq}(X)$ is a $\mbox{1-1}$ correspondence for $i>l$ and is onto for $i=l$.


equivariant maps, twisted coefficients

2010 Mathematics Subject Classification


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Published 30 November 2014