Edit: According to the essential comment of Alex Degtyarev, we revise the question as follows;

Assume that $\alpha$ and $\beta$ are two oriention preserving automorphism of Lie groups $O(n)$ and $O(m)$, respectively. Is there an oriention preserving automorphism of $O(n+m)$ which restriction to $O(n)\oplus O(m)$ equal to $\alpha \oplus \beta$?

One can repeat the same question for complex version,i.e:$U(n)$.

The reason for consideration of "oriention preserving" is that: It is unlike that the answer would be positive for $\alpha(z)=z$ and $\beta(w)=\bar{w}$, as automorphisms of $U(1)$.

The reason for consideration of $O(n)$, instead of $GL(n,\mathbb{R})$ is that: it is unlike that the answer would be positive for $\alpha(x)=x^{3},\;\beta(x)=x$, as automorphism of $GL(1,\mathbb{R})$.

Our main motivation:

We try to define a (possible) equivalent relation on Riemannian vector bundles as follows:

Assume that $E$ and $F$ are two vector bundles over $X$.

We say that $E$ is equivalent to $F$ if there are cocycles $g_{\alpha\beta}$ and $h_{\alpha\beta}$, respectively for $E$ and $F$ and an (oriention preserving) automorphism $\lambda$ of $O(n)$ such that $g_{\alpha\beta}=\lambda\circ h_{\alpha\beta}$.

Is this really an equivalent relation?(Transitive property?) Is there a sheaf theoretical language for this question(for this relation)?

Of course, every two ordinary isomorphic bundles, are equivalent in this definition.

If the answer is yes, We wish to define the direct product on this structure (on the space of all equivalent class of Riemannian vector bundles) to obtain a semigroup. So the above question is needed for "well define-ness". Then we consider the Grothendieck group of this semigroup. So we would have a functor $\tilde{\tilde{K}}(X)$.

Now perhaps a natural question is that what would be an appropriate version of Atiyah Janich theorem, here?

As Alex Degtyarev commented on previous version of this post, we ask that how can we define an appropriate higher order of $\tilde{\tilde{K}}$?

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    $\begingroup$ Are you sure that your equivalence relation is meaningful in the first place? E.g., what would happen over triple intersections? Then, to make a $K$-functor a cohomology theory, you'd need to prove some kind of Bott periodicity (or find another way to define $K^{>0}$. $\endgroup$ – Alex Degtyarev Jan 22 '15 at 10:43
  • $\begingroup$ @AlexDegtyarev For every two vector bundles, one can assume that they have the same system(open covering) of trivialization. So the definition of this relation is not meaning-less, but after your comment, I realize that the transitive property is not obvious.Can we prove transitive property. Regarding the second part of your comment, I am motivated by the following the equivalents class of vector bundles is in 1-1 correspondence with homotopy class of continuous maps from X to fredholm operatores. Now my question is that : With this new equivalent relation, what space play the same role $\endgroup$ – Ali Taghavi Jan 23 '15 at 12:27
  • $\begingroup$ as the role of fredholm operatores.So according to your comment I revise the question. Thank you for your comment. $\endgroup$ – Ali Taghavi Jan 23 '15 at 12:28
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    $\begingroup$ The automorphisms are inner, I think, so try inner automorphisms first. $\endgroup$ – Ben McKay Jan 23 '15 at 17:04
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    $\begingroup$ Your automorphism gives you the conjugate of any complex vector bundle. I think that every automorphism of the unitary group is either inner or the composition of this conjugation automorphism with an inner automorphism. $\endgroup$ – Ben McKay Jan 25 '15 at 9:27

EDIT: This is now a full answer, and the answer is "no". Let $G$ be a structure group as in the question. Let $\lambda$ and $\mu$ be automorphisms, then so is $\lambda\circ\mu$. We write $E\sim_\lambda F$ if $E$, $F$ have cocycles $g_{\alpha\beta}$, $h_{\alpha\beta}$ respectively such that $g_{\alpha\beta}=\lambda\circ h_{\alpha\beta}$. The following things are easy to check.

  1. $\sim_\lambda$ is compatible with the cocycle condition.

  2. If $g'_{\alpha\beta}=c_\alpha g_{\alpha\beta}c_\beta^{-1}$ is an equivalent cocycle for $E$, then the $0$-cochain $(\lambda\circ c_\alpha)$ can be used to find an equivalent cocycle $h'_{\alpha\beta}$ for $F$ to which $g'_{\alpha\beta}$ is again $\lambda$-related.

  3. $E\sim_\lambda F$ if and only if $F\sim_{\lambda^{-1}}E$.

  4. $E\sim_\lambda F$ and $F\sim_\mu D$ imply $E\sim_{\lambda\circ\mu}D$ (this needs step 2).

  5. If $\lambda$ is an inner automorphism induced by an element $g_0\in G$ and $g_{\alpha\beta}$ is $\lambda$-related to $h_{\alpha\beta}$, then $g_{\alpha\beta}$ and $h_{\alpha\beta}$ are in fact equivalent cocycles related by the constant $0$-cochain $g_0$.

Define $E\sim F$ if and only if there exist cocycles describing $E$ and $F$ which are $\lambda$-related for some $\lambda$. Then $E\sim F$ is the same as $E\cong F$ unless $G$ has outer automorphisms.

Outer automorphisms of simply connected, connected Lie groups can be read off from the Dynkin diagram. Neither $O(n)$ nor $U(n)$ is of this type, but you can still make some deductions.

For example, the central $S^1$ in $U(n)$ contains $n$ elements of $SU(n)$, which determines the action of outer automorphisms on the center for $n\ge 3$. Hence the group $U(n)$ has one (equivalence class of nontrivial) outer automorphisms given by complex conjugation (even for $n=1$, $2$). If $E\sim_\lambda F$, then $$c_1(E)=\pm c_1(F)$$ with "$-$" if and only if $\lambda$ is outer. Hence there is no way to extend identity on $U(m)$ and conjugation on $U(n)$ to an automorphism of $U(m+n)$ because in general $$c_1(E\oplus\bar F)\ne\pm(c_1(E)+c_1(F))\;.$$ So you cannot define your version of complex $K$-theory that way.

All outer automorphisms of $SO(n)$ come from inner automorphisms of $O(n)$ (the triality automorphisms of $Spin(8)$ do not act on $SO(8)$). The group $O(n)$ is generated by reflections, and each reflection is specified up to sign by its conjugation action on $SO(n)$ for $n\ge 3$. So the only possible outer automorphism of $O(n)$ is given by $\lambda(g)\mapsto g\cdot\det g$ in the case that $n$ is even (for $n=2$, this is actually an inner automorphism as well). If $E\sim_\lambda F$ then $F\cong E\otimes o(E)$, where $o(E)=\Lambda^nE$ denotes the orientation bundle. In particular, if $E$ is orientable, then $E\cong F$.

Using the splitting principle, one computes the Stiefel-Whitney classes $$w_k(E\otimes o(E))=\sum_{i=0}^k\binom{n-i}{k-i}w_i(E)\,w_1(E)^{k-i}\;.$$ Already for $n=4$ and $k=2$, one has $w_2(E\otimes o(E))=w_2(E)+w_1(E)^2$, so the bundles $E$ and $E\otimes o(E)$ are not stably isomorphic. To come back to the original question: for $m$ odd, the bundles $E\oplus\underline{\mathbb R}^m$ and $(E\otimes o(E))\oplus\underline{\mathbb R}^m$ are not diffeomorphic, and because $m+n$ is odd, there is no outer automorphism to relate them. So again, you cannot define a new $K$-group.


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