Questions regarding definitions of reduction of structure group of a Principal/Fiber/Vector bundle?

Question

Let $P,B$ be algebraic varieties and $G$ be an algebraic group.

Borel Construction:Let $p:P\rightarrow B$ be a principal $G$-bundle. Suppose $G$ acts on the left on a variety $F$. One can define a Borel Construction $P\times_G F$ to be the quotient of $P\times F/ \sim$ where $(x,f)\sim (xg,g^{-1}f)$. We have a natural projection from $q:P\times_G F\rightarrow X$ by defining $[x,f]\mapsto p(x)$.

Then $q:P\times_G F\rightarrow X$ is a fiber bundle over $B$ with fiber $F$ and structure group $G$ which has the same transition function.

Now let us start with a $q:Q\rightarrow B$ a principal $H$ bundle such that there is a morphism from $H\rightarrow G$. Take $F=H$, notice $H$ acts on $G$ from the left. Applying the Borel Construction we get $p:Q\times_H G\rightarrow X$ a fibre bundle with fiber $G$ and structure group $H$, via the morphism $H\rightarrow G $ we can think the transition functions taking value in $G$. Hence we get a Principal $G$ bundle.

Definition(Reduction of structure group): Given a principal $G$-bndle $p:E\rightarrow B$ we say that the structure group $G$ can be reduced to $H$ for some algebraic group $H$ with a morphism $H\rightarrow G$ if there exists a principal $H$-bundle $Q\rightarrow B$ and a $G$-equivariant morphism $\tau : Q\times_H G\rightarrow E$ so that the map is $\tau$ is $G$-equivariant.

• Question 1-Why don't we require $\tau$ to be an isomorphism?
• Question 2- Why cannot we simply say that "Given a principal $G$-bndle $p:E\rightarrow B$ we say that the structure group $G$ can be reduced to $H$ if the transition functions factor through a map $U_{\alpha}\cap U_{\beta}\rightarrow H$"
• Question 3- One usually defines reduction of a structure group of a fiber bundle by defining it to be the reduction of the structure of group of the associated Principal bundle. If the 'definition' in Question-2 is a valid one, then one can define reduction of a structure group of a fiber bundle similar to that, that is the transition functions (or may be another set of equivalent transition fucntion) factors through $H$.
• Question 4- Bott and Tu defines a vector bundle to have reduction of structure group to $H$, if there is a equivalent transition function which takes values at $H$. But if we think a vector bundle as a fibre bundle, we have another definition- 'The vector bundle is said to have a reduced structure groupp $H$ if the associated Principal Bundle have a reduced structure group H.' Are these two definitions equivalent?

Note: I am not demanding the action of the algebraic group $G$ on the fiber $F$ to be faithful. So that I don't have to necessarily to talk about reduction of structure group into a subgroup f $G$.

Answer 1

Question 1: Since the source and target are both principal $G$-bundles, a morphism, if it exists, will automatically be an isomorphism. So it's not necessary to impose it (although I agree it would be more transparent if it were included in the statement).

Question 2: You can, with the caveat that $H \to G$ may not be an inclusion, in which "factors through" has to be strengthened to "lifted to, in a way that they satisfy the cocycle condition".

E.g. giving a spin structure on an orientable Riemannian manifold corresponds to reducing the structure group $SO(n)$ on the bundle of oriented orthonormal frames to its spin double-cover. So here $H\to G$ is a double cover, not an inclusion.

Question 3: This is more-or-less correct, but there is the caveat that $H$ may not act faithfully on $X$ (since e.g. the map $H\to G$ may not be an inclusion, i.e. may have a kernel).

Question 4: Yes, the two notions are equivalent, if we assume that $H$ is a subgroup of $G$.