Let the group $G=\mathbb S^3$ act semi freely on a paracompact space $X$. Then exercise 12 of G.E. Breadon's book , Introduction to compact transformation groups pg 169 asks to derive the following exact sequence (Smith -Gyisn) with arbitrary coefficients.

.....$H^i(X^*,X^G)\rightarrow H^i(X)\rightarrow H^{i-3}(X^*,X^G)\oplus H^i(X^G)\rightarrow H^i(X^*,X^G)\rightarrow ....$, where $X^*$ denotes the orbit space, $X^G$ denotes the fixed point set and cohomology groups are cech cohomology groups.

-The exercise doesnt specify the maps involved in the above sequence?.

  • As in the case of smith gysin sequence of the circle bundle (sequence 10.5 page161 Introduction to compact transformation groups) is it true that here also the homomorphism $H^{i-3}(X^*, X^G)\rightarrow H^{i+1}(X^*,X^G)$ is the cup product with an element w, the generator of $H^4(X^*,X^G)$. ?

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