Mixing
Least number of local trivializations of a vector bundle
Let $X$ be a compact Hausdorff space, and let $pi: E to X$ be a real or complex vector bundle over $X$ of rank $r$. Then there exist finitely many local trivializations of $E$, i.e. there exists an open covering $(U_j)_{j=1}^k$ and homeomorphisms $h_j : pi^{-1}(U_j) to U_j times mathbb{K}^r$ such that $pi(h_j^{-1}(x,v)) = x$ for $j=1, ldots, k$.
Is there a name for the least such $k$, i.e. the least number of local trivializations required for the vector bundle? Also, are there some references looking into this concept?
algebraic-topology vector-bundles
asked Nov 21 '18 at 0:14
Ulrik
2,021917
add a comment |
Let $X$ be a compact Hausdorff space, and let $pi: E to X$ be a real or complex vector bundle over $X$ of rank $r$. Then there exist finitely many local trivializations of $E$, i.e. there exists an open covering $(U_j)_{j=1}^k$ and homeomorphisms $h_j : pi^{-1}(U_j) to U_j times mathbb{K}^r$ such that $pi(h_j^{-1}(x,v)) = x$ for $j=1, ldots, k$.
Is there a name for the least such $k$, i.e. the least number of local trivializations required for the vector bundle? Also, are there some references looking into this concept?
algebraic-topology vector-bundles
asked Nov 21 '18 at 0:14
Ulrik
2,021917
2
This MO question may be useful.
– Pedro Tamaroff♦
Nov 21 '18 at 11:55
2
Maybe this is mentioned in the linked MO question, but an early reference for this is James' "On category, in the sense of Lusternik-Schnirelman", where this quantity you're describing is denoted $operatorname{vecat}$ and compared to other notions of category.
– JHF
Nov 21 '18 at 16:06
add a comment |
Let $X$ be a compact Hausdorff space, and let $pi: E to X$ be a real or complex vector bundle over $X$ of rank $r$. Then there exist finitely many local trivializations of $E$, i.e. there exists an open covering $(U_j)_{j=1}^k$ and homeomorphisms $h_j : pi^{-1}(U_j) to U_j times mathbb{K}^r$ such that $pi(h_j^{-1}(x,v)) = x$ for $j=1, ldots, k$.
Is there a name for the least such $k$, i.e. the least number of local trivializations required for the vector bundle? Also, are there some references looking into this concept?
algebraic-topology vector-bundles
asked Nov 21 '18 at 0:14
Ulrik
2,021917
Let $X$ be a compact Hausdorff space, and let $pi: E to X$ be a real or complex vector bundle over $X$ of rank $r$. Then there exist finitely many local trivializations of $E$, i.e. there exists an open covering $(U_j)_{j=1}^k$ and homeomorphisms $h_j : pi^{-1}(U_j) to U_j times mathbb{K}^r$ such that $pi(h_j^{-1}(x,v)) = x$ for $j=1, ldots, k$.
Is there a name for the least such $k$, i.e. the least number of local trivializations required for the vector bundle? Also, are there some references looking into this concept?
algebraic-topology vector-bundles
algebraic-topology vector-bundles
asked Nov 21 '18 at 0:14
Ulrik
2,021917
asked Nov 21 '18 at 0:14
Ulrik
2,021917
asked Nov 21 '18 at 0:14
Ulrik
2,021917
asked Nov 21 '18 at 0:14
Ulrik
2,021917
asked Nov 21 '18 at 0:14
Ulrik
2,021917
2,021917
2
This MO question may be useful.
– Pedro Tamaroff♦
Nov 21 '18 at 11:55
2
Maybe this is mentioned in the linked MO question, but an early reference for this is James' "On category, in the sense of Lusternik-Schnirelman", where this quantity you're describing is denoted $operatorname{vecat}$ and compared to other notions of category.
– JHF
Nov 21 '18 at 16:06
add a comment |
2
This MO question may be useful.
– Pedro Tamaroff♦
Nov 21 '18 at 11:55
2
Maybe this is mentioned in the linked MO question, but an early reference for this is James' "On category, in the sense of Lusternik-Schnirelman", where this quantity you're describing is denoted $operatorname{vecat}$ and compared to other notions of category.
– JHF
Nov 21 '18 at 16:06
2
2
This MO question may be useful.
– Pedro Tamaroff♦
Nov 21 '18 at 11:55
This MO question may be useful.
– Pedro Tamaroff♦
Nov 21 '18 at 11:55
2
2
Maybe this is mentioned in the linked MO question, but an early reference for this is James' "On category, in the sense of Lusternik-Schnirelman", where this quantity you're describing is denoted $operatorname{vecat}$ and compared to other notions of category.
– JHF
Nov 21 '18 at 16:06
Maybe this is mentioned in the linked MO question, but an early reference for this is James' "On category, in the sense of Lusternik-Schnirelman", where this quantity you're describing is denoted $operatorname{vecat}$ and compared to other notions of category.
– JHF
Nov 21 '18 at 16:06
add a comment |
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2
This MO question may be useful.
– Pedro Tamaroff♦
Nov 21 '18 at 11:55
2
Maybe this is mentioned in the linked MO question, but an early reference for this is James' "On category, in the sense of Lusternik-Schnirelman", where this quantity you're describing is denoted $operatorname{vecat}$ and compared to other notions of category.
– JHF
Nov 21 '18 at 16:06