Two questions for vector bundles

[1591238460]

1 Let A → N, B → N be two vector bundles over a manifold N. How to show that there is a vector bundle Hom(A, B) whose fiber above x ∈ N is Hom(A, B)x := Hom(Ax, Bx)?
2 Let A → N, B → N be two vector bundles over a manifold N. Let C∞(A, B) denote the space of maps of vector bundles from A to B. How to show that C∞(A, B) is a C∞(N)-module?

Could you give me some ideas or reference ? Thank you!

 

[lavinia]

1 Let A → N, B → N be two vector bundles over a manifold N. How to show that there is a vector bundle Hom(A, B) whose fiber above x ∈ N is Hom(A, B)x := Hom(Ax, Bx)?

The bundle space,$E$, is just the collection of vector spaces, $Hom(A_x,B_x)$ with bundle projection, $π:E \rightarrow X$ defined by $π(L_x) = x$ where $L$ is a homomorphism in $Hom(A_x,B_x)$. You need to show that this space determines a vector bundle by showing that the coordinate functions for $A$ and $B$ determine coordinate functions for $Hom(A_x,B_x)$. This is not hard since the coordinate functions can be thought of as matrices that vary continuously over a coordinate neighborhood.( A coordinate neighborhood is an open set in the base above which the vector bundle is trivial.)

The topology of $E$ is determined from local trivializaions because the coordinate transformations connecting overlapping local trivializaions are continuous if the coordinate transformations for $A$ and $B$ are continuous. Generally speaking, the topology of a bundle is determined by coordinate transformations since it defines the "gluing" maps that tell one how to identify fibers above intersection points of coordinate neighborhoods. The topology of the bundle is the quotient topology of the space of local trivializations under these identifications.

In general operations on vector spaces such as tensor product, Hom, dual space, direct sum ... all determine new bundles from old.

2 Let A → N, B → N be two vector bundles over a manifold N. Let C∞(A, B) denote the space of maps of vector bundles from A to B. How to show that C∞(A, B) is a C∞(N)-module?

The smooth functions on $N$ is a ring under pointwise addition and multiplication. Pointwise multiplication of a bundle map by a smooth function on $N$ is again a bundle map.

 

[mathwonk]

lavinia, does this work to construct Hom(E,F)? looking at an n- bundle E as given by a one-cocyle with values in the linear automorphisms of R^n, we start with two such cocycles, one with values in GLn and one with values in GLm and we want a cocycle with values in Aut(Hom(R^n,R^m)). Just use conjugation, i.e. at a point x where the two cocycles have values A in GLn, and B in GLm, the new cocycle takes C in Hom(R^n,R^m), to A^-1.C.B.

 

[lavinia]

lavinia, does this work to construct Hom(E,F)? looking at an n- bundle E as given by a one-cocyle with values in the linear automorphisms of R^n, we start with two such cocycles, one with values in GLn and one with values in GLm and we want a cocycle with values in Aut(Hom(R^n,R^m)). Just use conjugation, i.e. at a point x where the two cocycles have values A in GLn, and B in GLm, the new cocycle takes C in Hom(R^n,R^m), to A^-1.C.B.

I think so. Just use the change of basis transformation.

$A^{-1}:R^n \rightarrow R^n$ followed by $C:R^n \rightarrow R^m$ then $B:R^m \rightarrow R^m$. This rewrites $C$ in the new bases for $R^n$ and $R^m$. This process is continuous in $A$ and $B$ so the gluing maps are continuous.

Milnor's Characteristic Classes defines the idea of "continuous functor" from products of vector spaces into vector spaces.. For the purposes of vector bundles the functors assumed to be covariant. Continuity means that the assigmenet of linear maps by the functor is continuous as a function of the linear maps. Each continuous functor of this type determines a new vector bundle from old ones. Continuity guarantees the continuity of the derived coordinate transformations. $Hom: V$ x $W \rightarrow Hom(V,W)$ is an example.

 

[mathwonk]

yes i got this idea from Atiyah's account in his book "K-theory" of Milnor's notion of continuous functors. i just tried to make it look shorter and more intrinsic, without picking bases for the spaces of matrices.

 

[lavinia]

yes i got this idea from Atiyah's account in his book "K-theory" of Milnor's notion of continuous functors. i just tried to make it look shorter and more intrinsic, without picking bases for the spaces of matrices.

Right. You don't need to actually choose a basis. A and B are just linear maps.

 

[mathwonk]

Actually, although I had read online that the idea of continuous functors was due to Milnor, it seems very clear, at least implicitly in the original book by Steenrod on fiber bundles. In the first 25 pages he describes bundles as given by a cocycle of coordinate transformations and shows that a continuous map between automorphism groups defines a new bundle, he calls a tensor bundle. he then gives the usual examples in the same way as we have discussed, but without using the word "functor". Atiyah's book appeared in 1964, and Milnor's in 1974, based on earlier notes from about 1958 as I recall. Still when Steenrod's book appeared in 1951, Milnor was only 20 years old and had apparently not begun to publish on the subject. That is, he had already proved at age 19 a significant theorem on knots, but at least his name is not in the bibliography of Steenrod's book on fiber bundles. So this construction seems to go back to the early days of the subject and only the terminology, but not the ideas, may be due later to Milnor or Atiyah. I.e. Milnor may have coined the term "continuous functor" but their use in constructing bundles apparently came earlier.

 

[lavinia]

Actually, although I had read online that the idea of continuous functors was due to Milnor, it seems very clear, at least implicitly in the original book by Steenrod on fiber bundles. In the first 25 pages he describes bundles as given by a cocycle of coordinate transformations and shows that a continuous map between automorphism groups defines a new bundle, he calls a tensor bundle. he then gives the usual examples in the same way as we have discussed, but without using the word "functor". Atiyah's book appeared in 1964, and Milnor's in 1974, based on earlier notes from about 1958 as I recall. Still when Steenrod's book appeared in 1951, Milnor was only 20 years old and had apparently not begun to publish on the subject. At least his name is not in the bibliography of Steenrod's book. So this construction seems to go back to the early days of the subject and only the terminology, but not the ideas, may be due to Milnor or Atiyah. I.e. Milnor may have coined the term "continuous functor" but their use in constructing bundles apparently came earlier.

Interesting. I believe that. Steenrod's book talks about derived bundles in detail

 

[mathwonk]

It is also interesting that the word "functor" does not occur in the index of Steenrod's 1951 book, although his book with Eilenberg which introduced that concept in connection with axiomatizing homology theory, appeared in 1952 and was apparently already completed in 1951.

 

[fresh_42]

Actually, although I had read online that the idea of continuous functors was due to Milnor, it seems very clear, at least implicitly in the original book by Steenrod on fiber bundles. In the first 25 pages he describes bundles as given by a cocycle of coordinate transformations and shows that a continuous map between automorphism groups defines a new bundle, he calls a tensor bundle. he then gives the usual examples in the same way as we have discussed, but without using the word "functor". Atiyah's book appeared in 1964, and Milnor's in 1974, based on earlier notes from about 1958 as I recall. Still when Steenrod's book appeared in 1951, Milnor was only 20 years old and had apparently not begun to publish on the subject. That is, he had already proved at age 19 a significant theorem on knots, but at least his name is not in the bibliography of Steenrod's book on fiber bundles. So this construction seems to go back to the early days of the subject and only the terminology, but not the ideas, may be due later to Milnor or Atiyah. I.e. Milnor may have coined the term "continuous functor" but their use in constructing bundles apparently came earlier.

Do you have some links or pdfs at hand where I can read about the functorial point of view?

 

[mathwonk]

no but a rank n real vector bundle is constructed by forming product bundles UjxR^n, on each set Uj of an open cover, and then gluing them on pairwise overlaps, hence one needs an invertible matrix at each point of UjmeetUk, hence a continuous function gjk from UjmeetUk to GL(n). And for compatibility of gluing one needs to have the cocycle condition, gij . gjk = gik. Then if one has a "continuous functor" from n diml spaces to m diml spaces, that means in particular one has a continuous map F from GL(n) to GL(m), i.e. a functor F not only sends each n diml vector space to an m diml one, but sends each endomorphism of n diml spaces to an endomorphism of m diml ones, preserving isomorphisms and compositions, and we ask this last map to be continuous.

then by simply composing, a cocycle gij defining a rank n bundle, yields (because F preserves compositions) another cocycle F(gij) defining a rank m bundle.

e.g. the wedge product of an n diml vector space with itself is a continuous functor and for each V we get a contin map from Aut(V) to Aut(wedge(r)V), and hence we can glue together the wedge products of the fibers of a bundle to get another bundle. You can do it too with several variables, and make a bundle out of the spaces Hom(V,W), opr VtensorW, or V*tensorW ≈ Hom(V,W), etc... I think if you google vector bundles this will be the only modern description out there.

here is a few words:

https://en.wikipedia.org/wiki/Smooth_functor

or page 6ff here:

ftp://http://www.mathematik.uni-kl.de/pub/scripts/wirthm/Top/vbkt_skript.pdf

 

[fresh_42]

no but a rank n real vector bundle is constructed by forming product bundles UjxR^n ...

Thank you.

 

[lavinia]

Do you have some links or pdfs at hand where I can read about the functorial point of view?

You can get Milnor's book Characteristic Classes on line as a PDF file.

 

[fresh_42]

You can get Milnor's book Characteristic Classes on line as a PDF file.

Thank you for the hint. Guess I have to change my printer cartridge.

 

[lavinia]

Thank you for the hint. Guess I have to change my printer cartridge.

Steenrod's book, The Topology of Fiber Bundles, if you haven't looked at it already, is one of the first that specifically focuses on fiber bundles. I find it difficult going but it covers other examples of bundles than vector bundles e.g. sphere bundles,covering spaces, and principal bundles. These are just as important as vector bundles. Milnor's book is advanced and his chapter on what vector bundles are ,while rigorous ,is also quite dense. Bott and Tu's Differential Forms in Algebraic Topology deals with bundles from the differentiable view point. This book is also advanced but gives a wonderful introduction to the use of calculus in algebraic topology. The book I started with is Singer and Thorpe's Book, lecture Notes on Elementary Topology and Geometry - also in PDF form. It deals with bundles in a way more elementary way but still has the modern viewpoint.

 

[fresh_42]

Steenrod's book, The Topology of Fiber Bundles, if you haven't looked at it already, is one of the first that specifically focuses on fiber bundles. I find it difficult going but it covers other examples of bundles than vector bundles e.g. sphere bundles,covering spaces, and principal bundles. These are just as important as vector bundles. Milnor's book is advanced and his chapter on what vector bundles are ,while rigorous ,is also quite dense. Bott and Tu's Differential Forms in Algebraic Topology deals with bundles from the differentiable view point. This book is also advanced but gives a wonderful introduction to the use of calculus in algebraic topology. The book I started with is Singer and Thorpe's Book, lecture Notes on Elementary Topology and Geometry - also in PDF form. It deals with bundles in a way more elementary way but still has the modern viewpoint.

I have a concrete question concerning tangent spaces. Therefore I appreciate all the hints you both have given to me and will certainly have at least a look at them, probably more than that. Thanks a lot.

 

[lavinia]

I have a concrete question concerning tangent spaces. Therefore I appreciate all the hints you both have given to me and will certainly have at least a look at them, probably more than that. Thanks a lot.

Ask the questions.

Mathwonk explained the idea of continuous functor .

 

[mathwonk]

what is your tangent space question? (or start a new thread? for it).

 

[fresh_42]

Ask the questions.

This would breach PF rules since there are no peer reviewed publications. It once arose from the investigation of Lie algebra algorithms, where there is a (I think published) thesis for $sl(2)$ in which case all becomes trivial. It gets interesting , e.g. the Heisenberg or Poincaré Algebra. But that's all unless meanwhile someone has had the same idea, which I don't know of. Edit: It's not the radical.