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Ali Taghavi
Ali Taghavi
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A kind of isomorphicity of vector bundles(A new kind of topological $k $theory)

Let $X$ be a connected topological space. Let $E$ be a $k$ dimensional sub vector bundle of the trivial vector bundle $X\times \mathbb{R}^n$. Then $E$ defines an idempotent with trace $k$ in $M_n(C(X))$. Conversely every trace $k$ idempotent of this algebra determines a $k$ dimensional sub bundle of $n$ domensional trivial bundle over $X$.

Two idempotents associated to two isomorphism bundles are Murray von Neumann equivalent.

Are there two non isomorphic $k$ dimensional sub bundle of $X\times \mathbb{R}^n$ for which their corresponding idempotents $e,f$ admit an automorphism $\alpha$ of $M_n(C(X))$ with $\alpha(e)=f$?

Note: The above question actualy defines an equivalent relation on the space of all $k$ dimensional subbundles of the $n$ dimensional trivial bundle.

A kind of isomorphicity of vector bundles(A new kind of topological $k $theory)

Let $X$ be a connected topological space. Let $E$ be a $k$ dimensional sub vector bundle of the trivial vector bundle $X\times \mathbb{R}^n$. Then $E$ defines an idempotent with trace $k$ in $M_n(C(X))$. Conversely every trace $k$ idempotent of this algebra determines a $k$ dimensional sub bundle of $n$ domensional trivial bundle over $X$.

Two idempotents associated to two isomorphism bundles are Murray von Neumann equivalent.

Are there two non isomorphic $k$ dimensional sub bundle of $X\times \mathbb{R}^n$ for which their corresponding idempotents $e,f$ admit an automorphism $\alpha$ of $M_n(C(X))$ with $\alpha(e)=f$?

A kind of isomorphicity of vector bundles

Let $X$ be a connected topological space. Let $E$ be a $k$ dimensional sub vector bundle of the trivial vector bundle $X\times \mathbb{R}^n$. Then $E$ defines an idempotent with trace $k$ in $M_n(C(X))$. Conversely every trace $k$ idempotent of this algebra determines a $k$ dimensional sub bundle of $n$ domensional trivial bundle over $X$.

Two idempotents associated to two isomorphism bundles are Murray von Neumann equivalent.

Are there two non isomorphic $k$ dimensional sub bundle of $X\times \mathbb{R}^n$ for which their corresponding idempotents $e,f$ admit an automorphism $\alpha$ of $M_n(C(X))$ with $\alpha(e)=f$?

Note: The above question actualy defines an equivalent relation on the space of all $k$ dimensional subbundles of the $n$ dimensional trivial bundle.

Source Link
Ali Taghavi
Ali Taghavi
  • 356
  • 8
  • 31
  • 123

A kind of isomorphicity of vector bundles(A new kind of topological $k $theory)

Let $X$ be a connected topological space. Let $E$ be a $k$ dimensional sub vector bundle of the trivial vector bundle $X\times \mathbb{R}^n$. Then $E$ defines an idempotent with trace $k$ in $M_n(C(X))$. Conversely every trace $k$ idempotent of this algebra determines a $k$ dimensional sub bundle of $n$ domensional trivial bundle over $X$.

Two idempotents associated to two isomorphism bundles are Murray von Neumann equivalent.

Are there two non isomorphic $k$ dimensional sub bundle of $X\times \mathbb{R}^n$ for which their corresponding idempotents $e,f$ admit an automorphism $\alpha$ of $M_n(C(X))$ with $\alpha(e)=f$?