## The reduction of the structute group of $\mathbb{R}^n$ -fiber bundles to a special subgroup of $Homeo(\mathbb{R}^n)$

Let $$G$$ be the group of all homeomorphisms $$f$$ of $$\mathbb{R}^n$$ which satisfy $$f(x+m)=f(x)+m,\quad \forall m\in \mathbb{Z}^n$$

In the other words, $$G$$ is the group of all equivariant homeomorphisms of Euclidean space with respect to the standard action of $$\mathbb{Z}^n$$ on $$\mathbb{R}^n$$.

Is there a fiber bundle whose fibers are homeomorphic to$$\mathbb{R}^n$$ but the structure group of the bundle can not be reduced to $$G$$? Is there a manifold $$M$$ such that the structure group of $$TM\to M$$, which is counted as a fiber bundle, can not be reduced to $$G$$?