Does $\mathfrak{N}_4$ contain at least four distinct elements?

Question

How do I see that the set $\mathfrak{N}_4$ consisting of all unoriented cobordism classes of smooth closed $4$-manifolds contains at least four distinct elements?

Answer 1

Thom proved that the unoriented bordism is a polynomial ring over $\mathbb{F}_2$ generated by elements $x_i$ with $i$ running over all numbers not of the form $2^k-1$. Thus, $\mathfrak{N}_4 = \mathbb{F}_2 \cdot \{x_2^2, x_4\}$. Thom moreover proves that the even $x_i$ can be represented by $\mathbb{RP}^i$.

Thus, our four manifolds are the empty manifold, $\mathbb{RP}^2\times \mathbb{RP}^2$, $\mathbb{RP}^4$ and the disjoint or connected sum of the latter two. Let's give a characteristic class proof that they are all different in $\mathfrak{N}_4$:

As Tsemo Aristide mentioned, Stiefel-Whitney numbers are cobordism invariants. Denoting the generator of $H^*(\mathbb{RP}^n;\mathbb{F}_2)$ by $u$, the total Stiefel-Whitney class is $(1+u)^{n+1}$. Thus, we get that $w_1(\mathbb{RP}^4) = u$ and $w_4(\mathbb{RP}^4) = u^4$ and all other Stiefel-Whitney classes are $0$. In particular, the non-trivial Stiefel-Whitney numbers are for the indices $(4,0,0,0)$ and $(0,0,0,1)$.

For $\mathbb{RP}^2\times \mathbb{RP}^2$ the total Stiefel-Whitney class is given by $(1+u+u^2)(1+v+v^2)$, where we denote the generator for $H^*(\mathbb{RP}^2;\mathbb{F}_2)$ of the second factor by $v$. The Stiefel-Whitney number for the index $(4,0,0,0)$ is zero because $w_1(\mathbb{RP}^2\times \mathbb{RP}^2) = u+v$ and $(u+v)^4 = 6u^2v^2 = 0$. On the other hand, $w_2 = u^2+uv+v^2$ and $w_2^2 = u^2v^2$ and thus the Stiefel-Whitney number with index $(0,2,0,0)$ is nonzero. In particular, these two manifolds have different Stiefel-Whitney numbers.

Let $M = \mathbb{RP}^2\times \mathbb{RP}^2 \coprod \mathbb{RP}^4$. Stiefel-Whitney numbers are additive under coproducts. Thus the Stiefel-Whitney numbers for the indices $(0,2,0,0)$ and $(4,0,0,0)$ are both $1$. Thus, they are different from the ones for the two manifolds above.

Answer 2

You have to find 4 manifolds with different stiefle-whitney numbers.