This is a question about signs.
Fix
- a based space $(X,x_0)$,
- a topological group $G$
- acting on $X$ from the left, so that the basepoint $x_0$ is fixed,
- a based map $\alpha\colon S^p\to G$ ($p\geq1$), and
- a based map $\beta\colon S^q\to X$ ($q\geq 1$).
I define (homotopy classes of) based maps $\gamma,\gamma'\colon S^{p+q}\to X$ as follows.
We have a fiber sequence $$ X\xrightarrow{i} X\times_G EG \to BG$$ with section $s\colon BG\to X\times_G EG$ determined by the basepoint $x_0$. Let $\gamma$ be the unique homotopy class such that $$ i\circ \gamma:=[s\circ \widetilde\alpha, i\circ \beta],$$ (Whitehead product) where $\widetilde\alpha\colon S^{p+1}\to BG$ is the adjoint to $\alpha$.
Consider the composite $$ S^p\times S^q \xrightarrow{\alpha\times\beta} G\times X\to X,$$ using the action $G\curvearrowright X$. Since $G$ fixes $x_0$, this composite factors through a map $$\kappa\colon H:=(S^p\times S^q)/(S^p\times *)\to X.$$ We define $$\gamma':= \kappa\circ \sigma,$$ where $\sigma\colon S^{p+q}\to H$ is a section-up-to-homotopy of the quotient map $H\to S^p\wedge S^q=S^{p+q}$. (It looks like $\sigma$ involves a choice, but in fact we can fix a "canonical" choice explicitly: it corresponds to the "obvious null-homotopy" of $[0,\iota_q]$. When $q\geq2$, we can fix $\sigma$ by requiring that the composite with the projection $H\to S^q$ be null.)
I can convince myself that $\gamma'\sim \gamma$. I can also convince myself that $\gamma'\sim -\gamma$. So which is it? Does the sign depend on $p$ or $q$?
Probably, you can't answer that without knowing about all the conventions I have used, that I haven't bothered to mention (or am completely oblivious to). The real question is: is there a reference that deals with this kind of formula, and is likely to be reliable about signs, and explicit about choices of convention which may affect signs? Alternately, do you know some example which clarifies how the signs should go?
