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For the First Chern class identity, use finite-dimensional classifying maps. A complex line bundle on a compact Hausdorff space embeds in a finite-dimensional trivial vector bundle, so it is a pullback vector bundle of the tautological bundle on some Complex projective space. For the two complex line bundles, the universal tensor product of vector bundles is classified by the mapIts restriction to either factor pulls the tautological bundle back to the tautological bundle on that factor. The degree-two cohomology of the product is the direct sum of the degree-two cohomology of its factors, so these two restrictions determine its First Chern class. Pulling back to $X$ provesThis proves the integral identity, including possible torsion elements in the cohomology of $X$; an argument using only curvature forms would not detect those torsion elements.

The splitting principle for complex vector bundles says that for a vector bundle $E\to X$ there is an iterated projective bundle $p:F(E)\to X$ for which $p^{\ast }$ is injective on integral cohomology, and $p^{\ast }E$ is a direct sum of complex line bundles. The same assertion holds with rational coefficients, and a common such space can split any finite collection of vector bundles. The injectivity follows at each stage from the projective bundle formula for complex vector bundles; a Hermitian metric splits the resulting filtration into a direct sum.

Write $x_i=c_1(L_i)$ for the Chern roots of the pulled-back vector bundle. Define the degree-$2j$ component of its Chern character byThe Fundamental theorem of symmetric polynomials expresses each power-sum symmetric polynomial ${\sum }_i{x}_i^j$ uniquely as an integral polynomial in the elementary symmetric polynomials, which are the Chern classes of $E$. Thus the definition makes sense on $X$ itself, without choosing actual Chern roots there. On a compact Hausdorff space, each vector bundle is classified by a finite-dimensional Grassmannian; its high-degree Chern character components therefore vanish. Hence this defines a class in the direct sum $H^{2\ast }(X;\mathbb{Q})$ even if $X$ has no finite dimension. On more general spaces an infinite Chern character is instead interpreted in the product of the even-degree cohomology groups.

On a common splitting space, the Chern roots of a direct sum are the combined root lists, whereas the First Chern class identity makes the Chern roots of a tensor product of vector bundles the pairwise sums $x_i+y_j$. ConsequentlyThe injectivity in the splitting principle for complex vector bundles descends both identities to $X$. Additivity extends the Chern character to the Grothendieck group by $\mathrm{ch}([E]-[F])=\mathrm{ch}(E)-\mathrm{ch}(F)$. Multiplicativity and $\mathrm{ch}(1)=1$ then show that the Chern character is a unital ring homomorphism.

For Complex K-theory of complex projective space, let $\gamma$ be the tautological bundle, put $t=1-[\gamma ]$, and set $h=-c_1(\gamma )$. The cohomology ring of complex projective space is $\mathbb{Z}[h]/(h^{n+1})$. The CW filtration gives a cofibrationThe K-theory six-term exact sequence and Complex K-theory of a sphere give, inductively,Here $b_n$ is the image of a Bott element from $S^{2n}$, normalized so that $\mathrm{ch}(b_n)=h^n$. This normalization follows from the multiplicative form of Bott periodicity: the Chern character of the degree-two Bott element is an integral generator, and multiplication yields the top-degree generator on $S^{2n}$. In particular, the abelian group $K^0({\mathbb{CP}}^n)$ is a free abelian group of rank $n+1$.

Assume inductively that $1,t,\dots ,t^{n-1}$ form an integral basis on ${\mathbb{CP}}^{n-1}$. Their lifts, together with $b_n$, form an integral basis on ${\mathbb{CP}}^n$. Sincethe Chern characters of this basis are linearly independent over $\mathbb{Q}$: their first nonzero degrees are $0,2,\dots ,2n$. Thus the Chern character is injective on $K^0({\mathbb{CP}}^n)$. By induction $t^n$ restricts to zero on ${\mathbb{CP}}^{n-1}$, so $t^n=a{b}_n$ for an integer $a$. Comparing Chern characters gives $a=1$. Also $\mathrm{ch}(t^{n+1})=0$, so $t^{n+1}=0$. This proves the integral answer, rather than merely its rationalization:The induction begins with ${\mathbb{CP}}^0$ a point and $t=0$.

For a based space $(X,x_0)$, the homotopy groupis the set of based homotopy classes, with its usual concatenation operation. A map $f:X\to Y$ is a weak homotopy equivalence when it induces a bijection on path components and an isomorphismfor every $k\ge 1$ and every basepoint $x$. It is an n-connected map when it is bijective on ${\pi }_k$ for $k<n$ and surjective on ${\pi }_n$; equivalently, every homotopy fiber is $(n-1)$-connected.

A CW complex is built from a discrete set of zero-cells by successively attaching $k$-discs along maps from their boundary spheres, with the weak topology and closure-finiteness conditions. Its filtration by skeleta is the CW filtration.

For any space $X$, form its singular simplicial set $\mathrm{Sing}X$. Its geometric realization of a simplicial set is a CW complex, with one cell for each nondegenerate singular simplex, and evaluation givesThe Simplicial approximation theorem identifies based maps and homotopies from finite simplicial spheres into $|\mathrm{Sing}X|$ with singular simplices in $X$. Consequently $ϵ$ induces a bijection on components and isomorphisms on all homotopy groups. Thus every space admits a CW approximation.

The vanishing assumptions do not permit removal of all $n$-cells. Take $n=2$ andThen ${\pi }_2(X)=0$, and homology of a finite cyclic group gives $H_2(X;\mathbb{Z})=0$. If a connected CW complex $Y$ had no two-cells, attaching cells of dimension at least three would not change the fundamental group of its one-skeleton. Hence ${\pi }_1(Y)$ would be a free group. A weak equivalence $Y\to X$ would instead give ${\pi }_1(Y)\cong \mathbb{Z}/r$, which is nontrivial and finite and therefore not free. No such $Y$ exists.

The Bott isomorphism is multiplication by the Bott element $\beta \in {\tilde{K}}^0(S^2)$:Together with the suspension isomorphism andit gives the Complex K-theory of a sphere

Letbe a CW filtration in which each quotient $Y_r/Y_{r-1}$ is a wedge of even-dimensional spheres. The six-term exact sequence in Topological K-theory, the sphere calculation, and induction giveand a short exact sequence whose new summand in $K^0(Y_r)$ is free abelian on the newly attached cells. Every such extension splits as an extension of free abelian groups, so $K^0(Y)$ is free, with one generator for each cell. This proves the Complex K-theory of an even-cell complex result.

The exterior product definesFor a point it is the identity. Attaching one layer of even cells gives corresponding exact sequences on the source and target; the sphere case is the suspension isomorphism, and induction with the Five lemma proves that the product map remains an isomorphism. This is the Künneth theorem for complex K-theory with an even-cell factor.

For a mapping torus $T_f$, the K-theory Wang sequence of a mapping torus containsWhen $K^{-1}(Z)=0$, exactness gives

For $Z={\mathbb{CP}}^2\times {\mathbb{CP}}^2$, the Complex K-theory of complex projective space and the K-theory Künneth isomorphism giveThe factor swap interchanges $x$ and $y$. Its invariant subgroup has the basisIt follows that the K-theory of the mapping torus of the factor swap on two complex projective planes is