Confused by a theorem in Milnor-Stasheff

• Apr 28, 2022

• #1

quasar987

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The k-th Pontrjagin class of a real vector bundle is defined as the 2k-Chern class of the complexified bundle. Therefor, a Pontrjagin class lives in cohomology with integer coefficients. But then the statement of Theorem 15.9 is that if the coefficient ring is taken to be a PID [itex]\Lambda[/itex] containing 1/2 (ex: Z[1/2] or Q, R, C), then the singular cohomology ring of a certain space G is the polynomial ring over [itex]\Lambda[/itex] in the Pontrjagin classes. But what is meant by a Pontrjagin class as an element of [itex]H^*(G,\Lambda)[/itex] ?? Is there a natural map [itex]H^*(G,\mathbb{Z})\rightarrow H^*(G,\Lambda)[/itex] that allows such an identification!?

 

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• Apr 28, 2022

• #2

quasar987

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Looking at it more closely it seems to me that the definition of the Euler class of an oriented bundle as an integer cohomology class would work just as well by replacing Z with any PID (we need a unit). And since the Chern classes are defined only in terms of the Euler class, then those as well are defined with any PID as coefficients, and so finally also for the Pontrjagin classes.

 

• Apr 28, 2022

• #3

mathwonk

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There is such a map as asked for in the last sentence of post #1. In general, cohomology is contravariant in the spaces and covariant in the coefficient modules, e.g. Thm. 3, page 237, of Algebraic Topology, by E. Spanier. The reason is that the Hom functor, used to turn chains into cochains, is covariant in the second variable, i.e. a map from A-->B, yields a map from maps X-->A to maps X-->B. And of course there is a unique ring map from the integers Z into to any ring with identity element.

 

• Apr 28, 2022

• #4

quasar987

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mathwonk said:

There is such a map as asked for in the last sentence of post #1. In general, cohomology is contravariant in the spaces and covariant in the coefficient modules, e.g. Thm. 3, page 237, of Algebraic Topology, by E. Spanier. The reason is that the Hom functor, used to turn chains into cochains, is covariant in the second variable, i.e. a map from A-->B, yields a map from maps X-->A to maps X-->B. And of course there is a unique ring map from the integers Z into to any ring with identity element.

Ahh, yep. Plus I knew of that construction from the all the theorems in Milnor-Stasheff of the form "The natural map H^*(X,Z)-->H^*(X,Z_2) sends such-and-such characteristic class to such-and-such Stiefel-Whitney class" but I didn't think of it. Instead I was trying to find the map in the Universal Coefficient Theorem for cohomology but couldn't find it there. Anyhow, thanks for the reply!

 

• May 5, 2022

• #5

mathwonk

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you are welcome. i also looked for it in the univ coefficient section, then kept looking when i didn't find!

 

1. What is the Milnor-Stasheff theorem?

The Milnor-Stasheff theorem, also known as the "loop space theorem," is a fundamental result in algebraic topology that provides a way to classify certain types of spaces called H-spaces. It states that any connected H-space is homotopy equivalent to a product of Eilenberg-Maclane spaces.

2. What is an H-space?

An H-space is a topological space with a multiplication operation that is associative and has a unit element. In other words, it is a space that behaves like a group, but may not necessarily have inverses for all elements.

3. How does the Milnor-Stasheff theorem relate to loop spaces?

The Milnor-Stasheff theorem shows that a connected H-space is homotopy equivalent to a loop space, which is the space of all continuous maps from a circle into the H-space. This connection between H-spaces and loop spaces is what gives the theorem its alternative name.

4. What is the significance of the Milnor-Stasheff theorem?

The Milnor-Stasheff theorem has many important applications in algebraic topology, including the classification of certain types of spaces and the study of homotopy groups. It also provides a powerful tool for understanding the structure of H-spaces and their relationship to loop spaces.

5. What are some key ideas in the proof of the Milnor-Stasheff theorem?

The proof of the Milnor-Stasheff theorem involves several key ideas, including the use of homotopy groups, the construction of a certain fibration, and the application of spectral sequences. It also relies on the concept of a Postnikov tower, which is a way of breaking down a space into simpler pieces that are easier to analyze.