Multiplicative distance - Revision history

en>David Eppstein: source

2014-05-04T21:41:02Z

source

← Older revision		Revision as of 23:41, 4 May 2014
Line 1:		Line 1:
	~~In [[topology]], a branch of mathematics, the '''clutching construction''' is a way of constructing fiber bundles, particularly vector bundles on spheres.~~		Friends call him Royal. Climbing is what adore performing. Bookkeeping is what I do for a residing. Some time in the past I selected to live in Arizona but I need to transfer for my family members.<br><br>Feel free to surf to my blog: extended auto warranty ([http://Marbellaclassified.com/2014/09/10/boost-your-auto-repair-knowledge-using-this-advice/ special info])

	~~==Definition==~~
	~~Consider the sphere <math>S^n</math> as the union of the upper and lower hemispheres <math>D^n_+</math> and <math>D^n_-</math> along their intersection, the equator, an <math>S^{n-1}</math>.~~

	Given trivialized [[fiber bundle]]s with fiber <math>F</math> and structure group <math>G</math> over the two disks, then given a map <math>f\colon S^{n-1} \to G</math> (called the ''clutching map''), glue the two trivial bundles together via ''f''.

	~~Formally, it~~ is the [[coequalizer]] of the inclusions <math>S^{n-1} \times F \to D^n_+ \times F \coprod D^n_- \times F</math> via <math>(x,v) \mapsto (x,v) \in D^n_+ \times F</math> and <math>(x,v) \mapsto (x,f(x)(v)) \in D^n_- \times F</math>: glue the two bundles together on the boundary, with a twist.

	~~Thus we have a map <math>\pi_{n-1} G \to \text{Fib}_F(S^n)</math>: clutching information on the equator yields a fiber bundle on the total space.~~

	~~In the case of vector bundles, this yields <math>\pi_{n-1} O(k) \to \text{Vect}_k(S^n)</math>, and indeed this map~~ is ~~an isomorphism (under connect sum of spheres on the right).~~

	~~===Generalization===~~
	The above can be generalized by replacing the disks and sphere with any closed triad <math>(X;A,B)</math>, that is, a space ''X'', together with two closed subsets ''A'' and ''B'' whose union is ''X''. Then a clutching map on <math>A \cap B</math> gives a vector bundle on ''X''.

	~~===Classifying map construction===~~

	~~Let <math>p : M \to N</math> be a fibre bundle with fibre <math>F</math>. Let <math>\mathcal U</math> be~~ a collection of pairs <math>(U_i,q_i)</math> such that <math>q_i : p^{-1}(U_i) \to N \times F</math> is a local trivialization of <math>p</math> over <math>U_i \subset N</math>. Moreover, we demand that the union of all the sets <math>U_i</math> is <math>N</math> (i.e. the collection is an atlas of trivializations <math>\coprod_i U_i = N</math>).

	~~Consider the space <math>\coprod_i U_i\times F</math> modulo the equivalence relation <math>(u_i,f_i)\~~in ~~U_i \times F</math> is equivalent to <math>(u_j,f_j)\in U_j \times F</math> if and only if <math>U_i \cap U_j \neq \phi</math> and <math>q_i \circ q_j^{-1}(u_j,f_j) = (u_i,f_i)</math>. By design,~~ the ~~local trivializations <math>q_i</math> give a fibrewise equivalence between this quotient space and the fibre bundle <math>p</math>.~~

	~~Consider the space <math>\coprod_i U_i\times Homeo(F)</math> modulo the equivalence relation <math>(u_i,h_i)\in U_i \times Homeo(F)</math> is equivalent~~ to ~~<math>(u_j,h_j)\~~in ~~U_j \times Homeo(F)</math> if and only if <math>U_i \cap U_j \neq \phi</math> and consider <math>q_i \circ q_j^{-1}</math>~~ to ~~be a map <math>q_i \circ q_j^{-1} : U_i \cap U_j \to Homeo(F)</math> then we demand that <math>q_i \circ q_j^{-1}(u_j)(h_j)=h_i</math>.~~
	Ie: in our re-construction of <math>p</math> we are replacing the fibre <math>F</math> by the topological group of homeomorphisms of the fibre, <math>Homeo(F)</math>. If the structure group of the bundle is known to reduce, you could replace <math>Homeo(F)</math> with the reduced structure group. ~~This is a bundle over <math>B</math> with fibre <math>Homeo(F)~~<~~/math~~> ~~and is a principal bundle. Denote it by~~ <~~math~~>~~p : M_p \~~to ~~N</math>. The relation~~ to ~~the previous bundle is induced from the principal bundle~~: ~~<math>~~(~~M_p \times F)/Homeo(F) = M</math>.~~

	So we have a principal bundle <math>Homeo(F) \to M_p \to N</math>. The theory of classifying spaces gives us an induced '''push-forward''' fibration <math>M_p \to N \to B(Homeo(F))</math> where <math>B(Homeo(F))</math> is the classifying space of <math>Homeo(F)</math>. Here is an outline:

	~~Given a <math>G<~~/~~math>-principal bundle <math>G \to M_p \to N<~~/~~math>, consider the space <math>M_p \times_{G} EG</math>~~. ~~This space is a fibration in two different ways:~~

	~~1) Project onto the first factor: <math>M_p \times_G EG \to M_p~~/~~G = N<~~/~~math>. The fibre in this case is <math>EG<~~/~~math>, which is a contractible space by the definition of a classifying space.~~

	~~2) Project onto the second factor: <math>M_p \times_G EG \to EG~~/~~G = BG</math>. The fibre in this case is <math>M_p</math>.~~

	Thus we have a fibration <math>M_p \to N \simeq M_p\times_G EG \to BG</math>. This map is called the '''classifying map''' of the fibre bundle <math>p : M \to N</math> since 1) the principal bundle <math>G \to M_p \to N</math> is the pull-~~back of the bundle <math>G \to EG \to BG</math> along the classifying map and 2) The bundle <math>p</math> is induced from the principal bundle as above.~~

	~~===Contrast with twisted spheres===~~
	~~{{see also\|Twisted sphere}}~~
	~~[[Twisted sphere]]s are sometimes referred to as a "clutching-type" construction, but this is misleading: the clutching construction is properly about fiber bundles.~~

	* In twisted spheres, you glue two ''disks'' along their boundary. The disks are a priori identified (with the standard disk), and points on the boundary sphere do not in general go to their corresponding points on the other boundary sphere. This is a map <math>S^{n-~~1} \to S^{n~~-~~1}</math>: the gluing is non~~-~~trivial in the base.~~
	* In the clutching construction, you glue two ''bundles'' together over the boundary of their base disks. The boundary spheres are glued together via the standard identification: each point goes to the corresponding one, but each fiber has a twist. This is a map <math>S^{n-~~1} \to G</math>: the gluing is trivial in the base, but not in the fibers.~~

	~~==References==~~
	* [[Allen Hatcher]]'s book-in-~~progress [http:~~/~~/www.math.cornell.edu/~hatcher/VBKT/VBpage.html Vector Bundles & K-Theory] version 2.0, p. 22.~~

	~~{{DEFAULTSORT:Clutching Construction}}~~
	~~[[Category:Topology]]~~
	~~[[Category:Geometric topology]]~~
	~~[[Category:Differential topology]]~~
	~~[[Category:Differential structures]~~]

en>Qetuth: more specific stub type

2012-01-01T11:18:15Z

more specific stub type

New page

In [[topology]], a branch of mathematics, the '''clutching construction''' is a way of constructing fiber bundles, particularly vector bundles on spheres.

==Definition==
Consider the sphere <math>S^n</math> as the union of the upper and lower hemispheres <math>D^n_+</math> and <math>D^n_-</math> along their intersection, the equator, an <math>S^{n-1}</math>.

Given trivialized [[fiber bundle]]s with fiber <math>F</math> and structure group <math>G</math> over the two disks, then given a map <math>f\colon S^{n-1} \to G</math> (called the ''clutching map''), glue the two trivial bundles together via ''f''.

Formally, it is the [[coequalizer]] of the inclusions <math>S^{n-1} \times F \to D^n_+ \times F \coprod D^n_- \times F</math> via <math>(x,v) \mapsto (x,v) \in D^n_+ \times F</math> and <math>(x,v) \mapsto (x,f(x)(v)) \in D^n_- \times F</math>: glue the two bundles together on the boundary, with a twist.

Thus we have a map <math>\pi_{n-1} G \to \text{Fib}_F(S^n)</math>: clutching information on the equator yields a fiber bundle on the total space.

In the case of vector bundles, this yields <math>\pi_{n-1} O(k) \to \text{Vect}_k(S^n)</math>, and indeed this map is an isomorphism (under connect sum of spheres on the right).

===Generalization===
The above can be generalized by replacing the disks and sphere with any closed triad <math>(X;A,B)</math>, that is, a space ''X'', together with two closed subsets ''A'' and ''B'' whose union is ''X''. Then a clutching map on <math>A \cap B</math> gives a vector bundle on ''X''.

===Classifying map construction===

Let <math>p : M \to N</math> be a fibre bundle with fibre <math>F</math>. Let <math>\mathcal U</math> be a collection of pairs <math>(U_i,q_i)</math> such that <math>q_i : p^{-1}(U_i) \to N \times F</math> is a local trivialization of <math>p</math> over <math>U_i \subset N</math>. Moreover, we demand that the union of all the sets <math>U_i</math> is <math>N</math> (i.e. the collection is an atlas of trivializations <math>\coprod_i U_i = N</math>).

Consider the space <math>\coprod_i U_i\times F</math> modulo the equivalence relation <math>(u_i,f_i)\in U_i \times F</math> is equivalent to <math>(u_j,f_j)\in U_j \times F</math> if and only if <math>U_i \cap U_j \neq \phi</math> and <math>q_i \circ q_j^{-1}(u_j,f_j) = (u_i,f_i)</math>. By design, the local trivializations <math>q_i</math> give a fibrewise equivalence between this quotient space and the fibre bundle <math>p</math>.

Consider the space <math>\coprod_i U_i\times Homeo(F)</math> modulo the equivalence relation <math>(u_i,h_i)\in U_i \times Homeo(F)</math> is equivalent to <math>(u_j,h_j)\in U_j \times Homeo(F)</math> if and only if <math>U_i \cap U_j \neq \phi</math> and consider <math>q_i \circ q_j^{-1}</math> to be a map <math>q_i \circ q_j^{-1} : U_i \cap U_j \to Homeo(F)</math> then we demand that <math>q_i \circ q_j^{-1}(u_j)(h_j)=h_i</math>.
Ie: in our re-construction of <math>p</math> we are replacing the fibre <math>F</math> by the topological group of homeomorphisms of the fibre, <math>Homeo(F)</math>. If the structure group of the bundle is known to reduce, you could replace <math>Homeo(F)</math> with the reduced structure group. This is a bundle over <math>B</math> with fibre <math>Homeo(F)</math> and is a principal bundle. Denote it by <math>p : M_p \to N</math>. The relation to the previous bundle is induced from the principal bundle: <math>(M_p \times F)/Homeo(F) = M</math>.

So we have a principal bundle <math>Homeo(F) \to M_p \to N</math>. The theory of classifying spaces gives us an induced '''push-forward''' fibration <math>M_p \to N \to B(Homeo(F))</math> where <math>B(Homeo(F))</math> is the classifying space of <math>Homeo(F)</math>. Here is an outline:

Given a <math>G</math>-principal bundle <math>G \to M_p \to N</math>, consider the space <math>M_p \times_{G} EG</math>. This space is a fibration in two different ways:

1) Project onto the first factor: <math>M_p \times_G EG \to M_p/G = N</math>. The fibre in this case is <math>EG</math>, which is a contractible space by the definition of a classifying space.

2) Project onto the second factor: <math>M_p \times_G EG \to EG/G = BG</math>. The fibre in this case is <math>M_p</math>.

Thus we have a fibration <math>M_p \to N \simeq M_p\times_G EG \to BG</math>. This map is called the '''classifying map''' of the fibre bundle <math>p : M \to N</math> since 1) the principal bundle <math>G \to M_p \to N</math> is the pull-back of the bundle <math>G \to EG \to BG</math> along the classifying map and 2) The bundle <math>p</math> is induced from the principal bundle as above.

===Contrast with twisted spheres===
{{see also|Twisted sphere}}
[[Twisted sphere]]s are sometimes referred to as a "clutching-type" construction, but this is misleading: the clutching construction is properly about fiber bundles.

* In twisted spheres, you glue two ''disks'' along their boundary. The disks are a priori identified (with the standard disk), and points on the boundary sphere do not in general go to their corresponding points on the other boundary sphere. This is a map <math>S^{n-1} \to S^{n-1}</math>: the gluing is non-trivial in the base.
* In the clutching construction, you glue two ''bundles'' together over the boundary of their base disks. The boundary spheres are glued together via the standard identification: each point goes to the corresponding one, but each fiber has a twist. This is a map <math>S^{n-1} \to G</math>: the gluing is trivial in the base, but not in the fibers.

==References==
* [[Allen Hatcher]]'s book-in-progress [http://www.math.cornell.edu/~hatcher/VBKT/VBpage.html Vector Bundles & K-Theory] version 2.0, p. 22.

{{DEFAULTSORT:Clutching Construction}}
[[Category:Topology]]
[[Category:Geometric topology]]
[[Category:Differential topology]]
[[Category:Differential structures]]