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If $M$ is a smooth manifold, the total Stiefel-Whitney class of $M$ is defined to by the total Stiefel-Whitney class of the tangent bundle, i.e. $w(M) := w(TM)$. If $M$ is a closed smooth manifold, t …

We have $H^*(BO(k); \mathbb{Z}_2) \cong \mathbb{Z}_2[w_1, \dots, w_k]$ where $\deg w_i = i$. In particular, $H^n(BO(k); \mathbb{Z}_2) \neq 0$ for every $n$ as $w_1^n$ is a non-zero element. Therefore …

As Mark Grant pointed out, there is no such example when $E$ is the tangent bundle of a smooth four-dimensional manifold because orientable smooth four-manifolds are spin${}^c$, so $W_3 =0$ and theref …

In your second edit, you ask whether there exists an example of such a bundle over a lower-dimensional manifold. Four-dimensional example Let $M = (\mathbb{RP}^2\times\mathbb{RP}^2)\#(S^1\times S^3) …

For $d \geq 4$, let $M_d = (S^1)^{d-4}\times\mathbb{CP}^2$. As tori are parallelisable, $w(M_d) = w(\mathbb{CP}^2)$, in particular $w_1(M_d) = 0$ and $w_2(M_d) \neq 0$, so $M_d$ does not admit a Spin, …

Recall that on a closed $n$-manifold $M$, there is a unique class $\nu_k$ such that $\operatorname{Sq}^k(x) = \nu_kx$ for all $x \in H^{n-k}(M; \mathbb{Z}_2)$; this is called the $k^{\text{th}}$ Wu cl …

Let $n$ be odd. Recall that $S^n$ is parallelisable if and only if $n = 1, 3, 7$. For every other $n$, there exists $k < n$ such that $S^n$ admits $k$ linearly independent vector fields, but not $k + …

Is there a smooth four-manifold $M$ such that $TM \cong \operatorname{End}(E)$ for some rank $2$ bundle $E \to M$? If $M$ is parallelisable, then one can take $E$ to be the trivial rank $2$ bundl …

Note, this answer was conceived before I understood Igor Belegradek's comments regarding the calculation of $p_1$. Suppose $E$ is orientable, i.e. $w_1(E) = 0$. Then $w_2(\operatorname{End}(E)) = w …

For any topological group $G$, there is a classifying space $BG$ and a principal $G$-bundle $EG \to BG$ called the universal principal $G$-bundle which is determined up to isomorphism by the fact that …

First note that the collection of orientable manifolds with $0 \neq w_2(M) = x^2$ for some $x \in H^1(M; \mathbb{Z}_2)$ is closed under products. Moreover, given two such manifolds of the same dimensi …

Let $i$ denote an immersion $N \to M$. There is an exact sequence of vector bundles on $N$ given by $$0 \to TN \to i^*TM \to \nu \to 0$$ where $\nu$ is the normal bundle. As total Stiefel-Whitney c …

A smooth manifold $M$ admits a pin$^+$ structure if and only if $w_2(M) = 0$, and a pin$^-$ structure if and only if $w_1(M)^2 + w_2(M) = 0$; see this page for some information on pin structures. The …

There is such an operation for $k = 2$ using virtual bundles. Note that $c_2(E\oplus F) = c_2(E) + c_1(E)c_1(F) + c_2(F)$ so \begin{align*} c_2(E) + c_2(F) &= c_2(E\oplus F) - c_1(E)c_1(F)\\ &= c …

The total Stiefel-Whitney class of any connected compact orientable surface $M$ can be computed fairly easily. As $M$ can be embedded in $\mathbb{R}^3$ with trivial normal bundle, $TM$ is stably trivi …