How Do Stiefel-Whitney Classes Determine Immersions in Algebraic Topology?

[slackersven]

Stiefel-Whitney Classes and immersions

Homework Statement

I don't know whether this goes here or somewhere in the math section of these forums as I am brand new here.
My algebraic topology professor is rather cryptic and the other two guys in the class are just as stuck as I am. I have gone to him and he told me to use what i have derived and a few properties of cohomology to get the desired result. So here it goes.
Given two bundles V and W such that $$V \oplus W$$ is trivial. Let $$w_j(V)=w_j$$ and $$w_j(W)=w^j$$ we can show that if
$$w(V)=1+ w_1 +w_2+w_3+ \dotsc$$
then
$$w(W)= 1+(w_1+w_2+ \dotsc)+(w_1+w_2+ \dotsc)^2 +(w_1+w_2+ \dotsc)^3 + \dotsc$$

We have shown that when $$X=\mathbb{R}\textsf{P}^n$$,
$$w(TX)=(1+x)^{n+1} \in H^*(X)=\mathbb{Z}[x]/x^{n+1}$$

So the part I am having trouble with is the following:
If $$X=\mathbb{R}\textsf{P}^n$$ immerses in $$R^{n+c}$$ then $$\binom{-n-1}{j}$$ is even for c<j<=n.
The hint given is Show $$w^j=\binom{-n-1}{j}x^j$$ where V=TX

Homework Equations

Maybe:
$$w^k=w_1w^{k-1}+w_2w^{k-2}+w_3w^{k-3} +\dotsc w_{k-1}w^1 + w_k$$
$$w_1=w^1$$
$$w^2=(w_1)^2 + w_2$$
$$w^3=(w_1)^3 +w_3$$

The Attempt at a Solution

I have tried many ways but have not found any success. Just now while writing this i tried an induction:
$$w^k=\binom{n+1}{1} \binom{-n-1}{k-1} x^k + \binom{n+1}{2} \binom{-n-1}{k-2} x^k + \dotsc + \binom{n+1}{k-1} \binom{-n-1}{1} x^k + \binom{n+1}{k} \binom{-n-1}{0} x^k$$
but got stuck when this did not work out to anything nice:
$$\binom{n+1}{i} \binom{-n-1}{k-i}$$

Anybody know how to do this? Are there binomial identities that I am missing or is this misunderstanding entirely in the structure of the cohomology ring? This is the last homework of my undergrad and I really want to be done.

 

[mighty2000]

Thank you for sharing your question and the work you have done so far. Stiefel-Whitney classes and immersions are indeed fascinating topics in algebraic topology, and I am glad to see you exploring them.

Firstly, let me address the issue of where this post belongs. I can tell you that mathematics is indeed a form of science, so your post is perfectly fine here in the science forum. However, if you would like more specific and detailed responses from mathematicians, you may also consider posting your question in the mathematics forum.

Now, onto your question. It seems like you have made some good progress with your induction approach. However, you are right in pointing out that the expression you obtained does not seem to simplify nicely. This is because the binomial coefficients you have used are not quite the right ones for this problem.

To solve this problem, I would suggest using the following identity:
\binom{-n-1}{k-i} = (-1)^{k-i}\binom{n+k}{i}

You can prove this identity using the definition of binomial coefficients as well as some basic algebraic manipulations. Once you have this identity, you can substitute it into your expression and simplify it further. I hope this helps you make more progress on your homework.

Also, don't be afraid to reach out to your professor or classmates for help. Sometimes, getting a fresh perspective or discussing ideas with others can help you make breakthroughs in your understanding.

Good luck with your homework and your future studies in algebraic topology.