Milnor Stasheff Problem

Question 1

Using a partition of unity, show that any vector bundle over a paracompact base space can be given a Euclidean metric.

Suppose $B$ is paracompact. Show that there is a correspondence between isomorphism classes of Eucldean vector bundle over $B$ and isomorphism classes of vector bundle over $B$. That we can forget or give Euclidean metric $g$. Moreover it is an isomorphism.

$$\left\{\text{Isomorphism Classes of Euclidean vector bundle over }B\right\}\leftrightarrow \left\{\text{Isomorphism Classes of vector bundle over }B\right\}$$

Solution

Let $\xi$ be a rank $n$ vector bundle with fiber $F$.
The bundle is locally diffeomorphic to $U\times F$.
Since it is trivial bundle locally, we have a local metric $g$ as a canonical Euclidean metric.
So we can choose orthonormal basis $\{e_i\}$ such that $g(e_i,e_j)={\delta }_{ij}.$

Since $B$ is paracompact, there is a partition of unity $\psi ={\sum }_{\text{finite}}{\psi }_i=1.$ Hence $\psi g$ gives a global metric that any vector bundle over a paracompact base space can be given a Euclidean metric.

Moreover, let $g_1$ and $g_2$ be Euclidean metrics on $\xi$.
Because the metric does not affect the rank of fibers, two fibers are isomorphic, so $(E,g_1)$ and $(E,g_2)$ are bundle isomorphic.
Therefore, there is a one-to-one correspondence between the collection of vector bundles with or without metrics.

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Question 2

If a vector bundle $\xi$ possesses a Euclidean metric, show that $\xi$ is isomorphic to its dual bundle $Hom(\xi ,{\epsilon }^1)$.

Solution

Let $F_b$ be a fiber of $\xi$ and $F_b^{\ast }$ be a fiber of $Hom(\xi ,{\epsilon }^1)$.

Pick a neighborhood $U$ of $b$.
Locally, since $\xi$ possesses metric $g$, similar to problem 1, we can choose orthonormal basis $\{e_i\}$.
And let $\{f^i\}$ be the basis of the vector bundle $Hom(\xi ,{\epsilon }^1)$ where $f^i$’s are dual basis of $e_i$.

We can construct the explicit vector space isomorphism through the metric $g$.
Let $g_{ij}=g(e_i,e_j)$.
Then the map ${\Phi }_b:F_b\to F_b^{\ast },e_i\mapsto \sum g_{ij}{f}^j$ gives a fiberwise isomorphism.

Since $g$ is a Euclidean metric that is continuous on the base space, $\xi$ and $Hom(\xi ,{\epsilon }^1)$ are isomorphic vector bundles.

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Question 3

Show that the Stiefel-Whitney classes of a Cartesian product are given by $w_k(\xi \times \eta )={\sum }_{i=0}^k{w}_i(\xi )\times w_{k-i}(\eta ).$

Solution

For simplicity, denote $\xi$ as the first bundle and $\eta$ as the second bundle.

Consider the two pullbacks $p{r}_i^{\ast }(E_i)$.
Let Whitney sum between two pullbacks $p{r}_1^{\ast }(E_1)\oplus p{r}_2^{\ast }(E_2)={\bigcup }_{x\in B}(x,F_x^1,F_x^2).$ Clearly, it is isomorphic to the product bundle $E_1\times E_2$.

By the axiom of Stiefel–Whitney classes, $w_k(\xi \times \eta )=w_k(p{r}_1^{\ast }{E}_1\oplus p{r}_2^{\ast }{E}_2)={\sum }_{i=0}^k{w}_i(p{r}_1^{\ast }(\xi ))\cup w_{k-i}(p{r}_2^{\ast }(\eta )).$

Also, by naturality and the product of cohomology classes, ${\sum }_{i=0}^{k}p{r}_1^{\ast }(w_i(\xi ))\cup p{r}_2^{\ast }(w_{k-i}(\eta ))={\sum }_{i=0}^k{w}_i(\xi )\times w_{k-i}(\eta ).$

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Question 4

Show that the set ${\mathfrak{N}}_n$ consisting of all unoriented cobordism classes of smooth closed $n$-manifolds can be made into an additive group.
This cobordism group ${\mathfrak{N}}_n$ is finite by corollary 4.11, and is clearly a module over $\mathbb{Z}/2$.
Using the manifolds ${\mathbb{RP}}^2\times {\mathbb{RP}}^2$ and ${\mathbb{RP}}^4$, show that ${\mathfrak{N}}_4$ contains at least four distinct elements.

Solution

Let $M$ and $N$ be $n$-manifolds.
Define the binary operation as $[M]+[N]=[M\bigsqcup N],$ which is clearly commutative.

• Well-defined: If $M\sim M^{\prime }$ and $N\sim N^{\prime }$ via cobordisms $W_M$ and $W_N$, then $M\bigsqcup N\sim M^{\prime }\bigsqcup N^{\prime }$ via $W_M\bigsqcup W_N$.
• Identity: The empty set $\varnothing$ is the identity element since $\varnothing =\partial S^{n+1}$.
• Inverse: Since $M\bigsqcup M$ bounds $M\times I$, we have $M\bigsqcup M\sim \varnothing$. Hence, $[M]=[M{]}^{-1}$.

Thus, ${\mathfrak{N}}_n$ is a $\mathbb{Z}/2$-module.
Since the number of Stiefel–Whitney numbers is finite, the order of ${\mathfrak{N}}_n$ is a finite group.

By problem 3, the following table is about Stiefel–Whitney classes of two manifolds: and $a\in H^1({\mathbb{RP}}^4;\mathbb{Z}/2)$, $b,c\in H^1({\mathbb{RP}}^2;\mathbb{Z}/2)$.

	$w_0$	$w_1$	$w_2$	$w_3$	$w_4$
${\mathbb{RP}}^4$	1	$a$	$a^2$	$a^3$	$a^4$
${\mathbb{RP}}^2\times {\mathbb{RP}}^2$	$1\times 1$	$c\times 1+1\times b$	$c^2\times 1+c\times b+1\times b^2$	$c^2\times b+c\times b^2$	$c^2\times b^2$

The set of partitions of 4 is $\Pi (4)=\{(0,0,0,1),(1,0,1,0),(0,2,0,0),(2,1,0,0),(4,0,0,0)\}.$

So the Stiefel–Whitney number table is:

Stiefel–Whitney Number	(0,0,0,1)	(1,0,1,0)	(0,2,0,0)	(2,1,0,0)	(4,0,0,0)
${\mathbb{RP}}^4$	1	1	1	1	1
${\mathbb{RP}}^2\times {\mathbb{RP}}^2$	1	0	1	0	0

Therefore, ${\mathfrak{N}}_4$ has at least four elements: $[\varnothing ],[{\mathbb{RP}}^4],[{\mathbb{RP}}^2\times {\mathbb{RP}}^2],[{\mathbb{RP}}^4]+[{\mathbb{RP}}^2\times {\mathbb{RP}}^2].$