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An immersion of a smooth $d$-manifold in ${\mathbb{R}}^{d+1}$ supplies a rank-one normal bundle $\nu$ with $TM\oplus \nu \cong {\epsilon }^{d+1}$. The Whitney product formula for Stiefel–Whitney classes therefore requires $w(\nu )=w(TM{)}^{-1}$. A real line bundle has no positive-degree classes beyond $w_1$, so every higher-degree part of the inverse total class must vanish. For ${\mathbb{RP}}^2\times {\mathbb{RP}}^2$, the stabilized tangent bundle of real projective space gives $w(TM{)}^{-1}=(1+a)(1+b)$ in ${\mathbb{F}}_2[a,b]/(a^3,b^3)$. Its nonzero degree-two part $ab$ excludes an immersion in ${\mathbb{R}}^5$ even though the normal line need not be orientable.

Let $Y\subseteq X$ be a properly embedded hypersurface. Choose local defining smooth functions $f_{\alpha }$ with nonzero differentials along $Y$. On overlaps they differ by nowhere-zero smooth factors. These transition factors define a real line bundle $L\to X$ and a transverse section $s$ whose zero set is $Y$.

The derivative of $s$ along its zero set is intrinsic and gives a surjective bundle mapIts kernel is $TY$, so it induces the vector bundle isomorphismThus the restriction of the defining line bundle is the normal bundle. These bundles have different original base spaces: $L$ lives over $X$, while $N_{Y/X}$ lives over $Y$. A global real-valued regular defining function would trivialize that restriction, explaining the obstruction from a nontrivial normal bundle.

An immersion $M^4\to {\mathbb{R}}^5$ would inject $TM$ into the trivial rank-five bundle. Taking its orthogonal complement gives a normal real line bundle $\nu$ with $TM\oplus \nu \cong {\epsilon }^5$. The Whitney product formula for Stiefel–Whitney classes requiresThe real line bundle need not be orientable, but it has no Stiefel–Whitney classes above degree one.

Let $a,b$ be the pulled-back degree-one generators from the two real projective planes. Their mod-two cohomology ring, by the Künneth theorem, is ${\mathbb{F}}_2[a,b]/(a^3,b^3)$. To compute the tangent classes, at a line $L$ the projective tangent space is $\mathrm{Hom}(L,L^{\perp })$. Adding $\mathrm{Hom}(L,L)\cong \mathbb{R}$ gives $\mathrm{Hom}(L,{\mathbb{R}}^3)\cong 3L$, with the real line identified with its dual using a metric. Thus the stabilized tangent bundle of real projective space gives $T{\mathbb{RP}}^2\oplus {\epsilon }^1\cong 3\gamma$ andThe tangent bundle of the product is the direct sum of the pulled-back tangent bundles. Since $(1+a+a^2)(1+a)=1$ in this truncated ring, and similarly for $b$,The degree-two class $ab$ is nonzero by the product-ring basis. It cannot be a class of the rank-one bundle $\nu$. Equivalently $w_1(TM{)}^2+w_2(TM)=ab\ne 0$, violating the codimension-one Stiefel–Whitney immersion obstruction. Hence ${\mathbb{RP}}^2\times {\mathbb{RP}}^2$ admits no immersion in ${\mathbb{R}}^5$.

Let $E$ have rank $r\ge 1$, let $p:\mathbb{P}(E)\to X$ be its projectivization of a real vector bundle, and let $\lambda \subset p^{\ast }E$ be its tautological bundle. Put $u=w_1(\lambda )\in H^1(\mathbb{P}(E);{\mathbb{F}}_2)$. The Projective bundle definition of Stiefel–Whitney classes starts from the isomorphismIn particular, $p^{\ast }$ is injective. Expand $u^r$ uniquely in this basis. The coefficients define the Stiefel–Whitney classes through the relationSet $w_0(E)=1$ and $w_i(E)=0$ for $i>r$; for a rank-zero vector bundle set $w(E)=1$. Uniqueness also proves naturality of the Stiefel–Whitney classes under a pullback vector bundle.

To prove the Whitney product formula for Stiefel–Whitney classes, apply the splitting principle for real vector bundles simultaneously to $E$ and $E^{\prime }$. Its injective pullback on mod-two cohomology follows by iterating the Projective bundle definition of Stiefel–Whitney classes. It suffices to compute for a split vector bundle $E={⨁}_{i=1}^r{L}_i$. Put $x_i=w_1(L_i)$.

On $\mathbb{P}(E)$, the coordinate projections of the tautological inclusion $\lambda \hookrightarrow p^{\ast }E$ give sections of the real line bundles ${\lambda }^{\ast }\otimes p^{\ast }{L}_i$ with no common zero. The mod-two Euler class of a real line bundle is its first Stiefel–Whitney class, and the First Stiefel–Whitney class of a tensor product of real line bundles gives $u+p^{\ast }{x}_i$. Multiplication of Euler classes under a direct sum, and vanishing of the Euler class of a vector bundle with a nowhere-zero section, therefore giveComparing with the unique monic relation defining the Stiefel–Whitney classes shows that $w_j(E)$ is the $j$th elementary symmetric polynomial in the $x_i$. Thus $w(E)={\prod }_i(1+x_i)$. A direct sum joins the two lists, so injectivity of the splitting pullback gives

For the immersion obstruction, use the notation $g=(f,f)$: the printed $f\times f$ here means the map from one copy of Real projective space into the product. Write $x=w_1({\gamma }_{\mathbb{R}})$ and let $z$ be the mod-two reduction of $c_1({\gamma }_{\mathbb{C}}^{\ast })$. The mod-two cohomology ring of real projective space and the cohomology ring of complex projective space giveNontriviality in degree two forces $f^{\ast }z=x^2$. Necessarily $k\ge 1$, since the degree-two cohomology of ${\mathbb{CP}}^0$ vanishes.

The standard tangent bundle identities for Real projective space and Complex projective space areFor the underlying real vector bundle of a complex line bundle, its total Stiefel–Whitney class is $1+c_1\mathrm{mod}2$. HenceIf $g$ is homotopic to an immersion $j$, its normal bundle $\nu$ has rank $4k-(2k+1)=2k-1$, andThe Whitney product formula for Stiefel–Whitney classes and homotopy invariance of cohomology yieldThe division is legitimate in the truncated cohomology ring, since $1+x$ is a unit. The degree-$2k$ component isIt must vanish because $2k>\mathrm{rank}\nu$, while $x^{2k}\ne 0$. Therefore $k$ must be odd.

Choose a locally finite cover $(U_{\alpha })$ such that on every chart meeting $Y$ there is a smooth defining function $f_{\alpha }$ withand take $f_{\alpha }=1$ on charts disjoint from $Y$. On an overlap, the supplied division lemma extendssmoothly across $Y$. After shrinking the charts, this extension is nowhere zero. The identities $g_{\alpha \beta }{g}_{\beta \gamma }=g_{\alpha \gamma }$ make these functions transition functions for a real line bundle $L\to X$.

Choose local frames $e_{\alpha }$ with $e_{\beta }=g_{\alpha \beta }{e}_{\alpha }$. Then the local sectionsagree on overlaps and define a global section. Its zero set is exactly $Y$. Along $Y$, its vertical derivative is represented by the nonzero covector $d{f}_{\alpha }$, so $s$ is transverse to the zero section, as in the transverse intersection theorem. This is the defining line bundle of a properly embedded hypersurface.

A real line bundle is a rank-one real vector bundle. Over a paracompact Hausdorff space, or over a smooth manifold in the smooth category, it is orientable exactly when it is a trivial vector bundle.

To prove the nontrivial implication, choose a fiber metric. An orientation of a vector bundle selects one of the two unit vectors in each fiber. These choices agree continuously in oriented local frames, giving a global nowhere-zero section. The map $(x,t)\mapsto ts(x)$ is then a fiber-linear product isomorphism. Conversely, a trivial vector bundle has the positive standard basis as a continuous orientation of a vector bundle.

The Möbius line bundle over a circle is the basic counterexample to triviality: one circuit reverses the fiber orientation, so no global positive unit section of a vector bundle can be chosen.

Over Real projective space ${\mathbb{RP}}^n$, the real tautological line bundle isOn the standard chart $U_i=\{[x]:x_i\ne 0\}$, let $s_i([x])=x/x_i$. A local trivialization sends $([x],t)$ to $([x],t{s}_i([x]))$, with inverse $([x],v)\mapsto ([x],v_i)$. These maps prove that this is a real line bundle. Its unit sphere bundle is the antipodal map quotient $S^n\to {\mathbb{RP}}^n$. Its mod-two Euler class of a real line bundle is the degree-one generator of the mod-two cohomology ring of real projective space.

The Stiefel–Whitney classes of a real rank-$r$ vector bundle $E\to X$ are characteristic cohomology classesThey are natural under pullback vector bundles and obey the Whitney product formula for Stiefel–Whitney classes, $w(E\oplus F)=w(E)w(F)$. For a real line bundle $L$, the normalization is $w(L)=1+w_1(L)$, with $w_1$ the class classifying its orientation double cover. These properties determine the classes through the splitting principle for real vector bundles.

The first Stiefel–Whitney class vanishes exactly when the vector bundle is orientable. The top Stiefel–Whitney class is its Euler class with mod-two coefficients and obstructs a nowhere-zero section. A trivial vector bundle has total class one; the Möbius line bundle has nonzero $w_1$.

Vanishing of every positive-degree Stiefel–Whitney class need not imply triviality. The tangent bundle of the two-dimensional sphere has $w_1=0$ and $w_2=\chi (S^2)\mathrm{mod}2=0$, although its integral Euler class is nonzero. The classes detect useful obstructions while forgetting information lost by reduction modulo two.