A simply connected CW complex of finite type has finitely generated homotopy groups in every degree. More generally, finite generation of its integral homology groups in each degree suffices. This is a finiteness theorem, not a claim that the groups are finite. A finite fundamental group allows the same conclusion after passing to a universal cover with finite type: a finite number of base cells lifts to a finite number of covering cells in each degree. It permits higher homotopy to be killed with finitely many cell attachments per dimension.

The universal cover of $S^1$ gives the first case. The Hopf fibration $S^1\to S^3\to S^2$ identifies ${\pi }_3(S^2)$ with ${\pi }_3(S^3)=\mathbb{Z}$. Starting from ${\pi }_4(S^3)=\mathbb{Z}/2$, the Freudenthal suspension theorem makes every subsequent suspension in these degrees an isomorphism. The stable range does not include the first step from ${\pi }_3(S^2)$ to ${\pi }_4(S^3)$, so the integer group at $n=2$ is not a contradiction.

A Postnikov tower of a connected CW complex $X$ consists of spaces $P_{n}X$ with ${\pi }_i(P_{n}X)={\pi }_i(X)$ for $i\le n$ and zero for $i>n$, together with compatible maps $X\to P_{n}X\to P_{n-1}X$. For simply connected $X$, each stage is a principal Eilenberg–MacLane fibration with fibre $K({\pi }_{n}X,n)$, classified by a Postnikov invariant. The canonical map $X\to {\mathrm{holim}}_n{P}_{n}X$ is a weak equivalence: in each homotopy degree the tower is eventually constant, so the derived inverse-limit obstruction vanishes.

The geometric Whitehead product of based maps $\alpha :S^p\to Y$, $\beta :S^q\to Y$ is their wedge sum map composed with the attaching map $S^{p+q-1}\to S^p\vee S^q$ of the top cell of $S^p\times S^q$. With the usual geometric signs it satisfies $[\alpha ,\beta {]}_W=(-1{)}^{pq}[\beta ,\alpha {]}_W$. Regrading ${\pi }_{p}Y$ to degree $p-1$ and setting $\{\alpha ,\beta \}=(-1{)}^p[\alpha ,\beta {]}_W$ gives the graded Lie bracket transported from the Samelson product on the loop space. In degrees $p,q\ge 2$ it is bilinear; fundamental-group inputs instead record commutators and the action of ${\pi }_1$ on higher homotopy.

Let $PX$ consist of paths starting at the chosen basepoint of $X$. Endpoint evaluation is a fibration $\Omega X\to PX\to X$, and $PX$ is contractible by shortening each path. Pulling it back along a map $Y\to X$ gives its homotopy fibre, after reversing the chosen path direction if needed. This makes an acyclic relative Sullivan algebra modelling $PX$ useful for explicit rational fibre calculations.

A principal fibration, in the homotopy-theoretic sense, is a homotopy pullback of a path-space fibration $PC\to C$ along a map $k:B\to C$. Its fibre is $\Omega C$ and acts on the fibre paths by concatenation, freely and transitively up to the relevant homotopies. Principal bundles for a topological group give this construction through its classifying space. When $C=K(A,n+1)$, the fibre is $K(A,n)$ and the principal fibration is classified by $H^{n+1}(B;A)$, yielding a principal Eilenberg–MacLane fibration.

If $f:X\to Y$ has maps $g,h:Y\to X$ with $fg\simeq 1_Y$ and $hf\simeq 1_X$, then $h\simeq hfg\simeq g$, so $g$ is a two-sided homotopy inverse. More generally, if $fg$ and $hf$ are homotopy equivalences, composing $g$ and $h$ with their respective homotopy inverses supplies a right and a left homotopy inverse of $f$, reducing to the first case.

A homotopy equivalence of a closed surface cannot carry a simple closed curve of nonzero integral homology class onto a contractible simple closed curve. Any map into the latter factors through a circle whose inclusion induces zero on first homology; the former class would be killed by an isomorphism. Such a carrying map can nevertheless exist: project a handle generator onto a circle and compose with the contractible embedded circle.

For $D={\mathrm{hocolim}}_j{T}_j$, applying $[-,E{]}_{\mathrm{st}}$ to the telescope cofiber sequence of spectra givesHere ${\lim }^1$ is the cokernel of $1-r$ on the product of the inverse tower, where $r$ uses its bonding maps. This records the obstruction to uniqueness when compatible stage maps are realized.

For a contravariant pointed functor on based CW homotopy types, half-exactness on a cofibration means that $F(X/A)\to F(X)\to F(A)$ is exact at $F(X)$: the image is the inverse image of the distinguished point. This does not demand injectivity or surjectivity at the ends. Set-valued exactness and group-valued exactness must be distinguished.

A based topological space with a continuous map $m$ whose left and right restrictions at the basepoint are homotopic to the identity. Associativity and inverses up to homotopy are additional conditions, not part of the bare definition.

For maps $A\to X$ and $A\to Y$, the double mapping cylinder $X\bigsqcup (A\times I)\bigsqcup Y$, with each endpoint attached by its given map, models the homotopy pushout. A map out of this model is a pair of maps on $X$ and $Y$ together with a homotopy between their restrictions to $A$. This is why compatible homotopy classes extend over a homotopy pushout. For a CW pair, the inclusion is a cofibration and the ordinary pushout gives the same homotopy type.

For the standard coordinate inclusion of spheres, write points as $(u,v)\in {\mathbb{R}}^{m+1}\times {\mathbb{R}}^{n-m}$, with $v\ne 0$ in the complement. The map $(u,v)\mapsto (u,v/\Vert v\Vert )$ is a homeomorphism to $B_{\mathrm{open}}^{m+1}\times S^{n-m-1}$, with inverse $(u,w)\mapsto (u,\sqrt{1-\Vert u{\Vert }^2}w)$. Contracting the ball factor proves the displayed homotopy equivalence. In codimension one there are two contractible components, so integral $H_0$ is ${\mathbb{Z}}^2$; in higher codimension the only nonzero homology groups are $H_0=\mathbb{Z}$ and $H_{n-m-1}=\mathbb{Z}$.

Give a leaf rank zero, and use Hessenberg natural sum over the immediate rooted subtrees. Every rank is below epsilon zero; conversely repeated exponentiation and finite summation encode every Cantor normal form below it. A rooted-tree homeomorphic embedding implies nondecrease of the rank. When roots match, child subtrees embed into distinct branches and exponentiation and natural sum preserve their inequalities. When the source root maps below the host root, its rank is bounded by that of a proper subtree, strictly below the host rank.

The classical mutual information obtained by any measurement of a quantum ensemble is at most its Holevo quantity. This follows from the data-processing inequality for quantum relative entropy applied to the measurement channel.

For equiprobable qubit states with Bloch vectors $r,s$, their density-matrix difference is $(r-s)\cdot \sigma /2$, with eigenvalues $\pm |r-s|/2$. An effect $0\le E\le I$ declaring the first state yields success $1/2+\mathrm{Tr}[E({\rho }_r-{\rho }_s)]/2$. Selecting its positive-eigenvalue eigenspace maximizes this trace and proves the displayed result. Measure along $r-s$ and attach the corresponding state labels; if the vectors coincide, no measurement improves on guessing.

A decreasing filtration defines a Hodge structure when $H_{\mathbb{C}}=F^p\oplus \overline{F^{w-p+1}}$ for every $p$. The decomposition is recovered as $H^{p,q}=F^p\cap \overline{F^q}$. This filtration is the holomorphic datum that varies in a variation of Hodge structure.

A polarization is a nondegenerate integral bilinear form of symmetry $(-1{)}^w$, orthogonal on Hodge summands except complementary bidegrees, satisfying the displayed positivity for every nonzero $v\in H^{p,q}$. This convention takes the cup-product pairing as the primitive geometric polarization. Omitting the constant sign factor and rescaling $Q$ is an equivalent convention. The full intersection pairing of a K3 surface is not itself a polarization: its real $(1,1)$ space has one positive direction as well as nineteen negative directions.

A pure integral variation consists of a local system of free abelian groups, holomorphic filtration subbundles of its associated flat bundle, and fibrewise Hodge structures of fixed weight and Hodge numbers. Its flat connection satisfies Griffiths transversality. A polarized variation additionally has a flat integral bilinear form polarizing every fibre.

This Poisson equation uses the Hodge Laplacian on differential forms. On a closed Riemannian manifold, the obstruction to solving it is the component of the source along harmonic differential forms. The Green operator of the Hodge Laplacian supplies a solution when that component vanishes.