Homotopy group (${\pi }_n(X,x_0)$)

The $n$th homotopy group consists of based homotopy classes of maps $S^n\to X$. It is a group for $n\ge 1$ and an abelian group for $n\ge 2$.

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.

The Whitehead square of a class $\alpha \in {\pi }_{m}Y$ is $[\alpha ,\alpha ]\in {\pi }_{2m-1}Y$. For the identity class of an even-dimensional sphere, the Hopf invariant is two with the standard product-cell orientation: folding the two sphere summands makes the middle cohomology class of the mapping cone pull back to $a+b$ on $S^m\times S^m$, and $(a+b{)}^2=2ab$. For odd $m$, graded symmetry makes the square two-torsion, hence rationally zero.

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.

For an abelian group $A$, a principal fibration with fibre $K(A,n)$ is a homotopy pullback of the path fibration $PK(A,n+1)\to K(A,n+1)$. It is therefore the homotopy fibre of a map $k:B\to K(A,n+1)$ and is classified by $[k]\in H^{n+1}(B;A)$. The fibre acts by translation up to homotopy; in this untwisted description the fundamental group acts trivially on $A$. This is the principal-fibration form used for simply connected Postnikov towers.

The $n$th Postnikov invariant is the cohomology class $k_n\in H^{n+1}(P_{n-1}X;{\pi }_{n}X)$ whose representing map has homotopy fibre $P_{n}X$. It specifies how the new homotopy group is attached to the previous stage. A zero invariant gives the product $P_{n}X\simeq P_{n-1}X\times K({\pi }_{n}X,n)$. Merely tensoring homotopy groups with $\mathbb{Q}$ does not describe rationalization of a topological space unless these classes are also transported.

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 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.

Based relative maps of a disk into $X$ with its boundary mapped into $A$, modulo homotopies preserving these conditions. Replacing a map by its mapping cylinder gives relative groups for a general map. They fit into the long exact sequence of relative homotopy groups and detect connectivity of the map.

For a based CW triad with connected intersection $A$, if $(B,A)$ is $p$-connected and $(C,A)$ is $q$-connected, $p,q\ge 1$, the comparison map is an isomorphism for $i<p+q$ and a surjection for $i=p+q$. This connectivity form of the Blakers-Massey theorem controls the failure of ordinary relative homotopy to satisfy unrestricted excision.

Inclusion, the map sending an absolute disk class to a relative one, and restriction to the disk boundary give this exact sequence. Its low-dimensional end is interpreted with pointed sets and the usual fundamental-group action. For a mapping-cylinder pair it measures the failure of a map to induce homotopy isomorphisms.

A connected Eilenberg–MacLane space $K(G,n)$, $n\ge 1$, has ${\pi }_n\cong G$ and all other positive-degree homotopy groups trivial. The group $G$ must be abelian when $n\ge 2$, but may be nonabelian when $n=1$. For an abelian group, the Dold–Kan correspondence constructs a simplicial model from the chain complex concentrated in degree $n$. A classifying space of a discrete group gives the general $K(G,1)$ case.

For $n\ge 4$, the first four possible degrees have integral groups $\mathbb{Z}/2,0,\mathbb{Z}/2,\mathbb{Z}/2$ in degrees $n,n+1,n+2,n+3$. Mod-two Betti numbers count cyclic summands, while the nonzero first Bockstein actions on $u,{\mathrm{Sq}}^{2}u,{\mathrm{Sq}}^2{\mathrm{Sq}}^{1}u$ show that their orders are two. This prevents mistaking a nonzero mod-two group for a higher-order integral cyclic group.

For $n\ge 1$, the mod-two cohomology of $K(\mathbb{Z}/2,n)$ is polynomial on the admissible operations in the displayed strict-excess range, including the empty operation. It is obtained from path-space transgressions and compatibility with Steenrod squares. Products must still be included when listing low-degree vector-space bases.

For $n\ge 2$, the positive-degree dimensions in degrees $n,n+1,n+2,n+3$ are $1,1,1,2$. The first three classes are $u,{\mathrm{Sq}}^{1}u,{\mathrm{Sq}}^{2}u$. In degree $n+3$, use ${\mathrm{Sq}}^{3}u$ and ${\mathrm{Sq}}^2{\mathrm{Sq}}^{1}u$ for $n\ge 3$; at $n=2$ replace the unstable first class by $u{\mathrm{Sq}}^{1}u$. At $n=2$ the degree-four class is $u^2$, and at $n=3$ the degree-six class ${\mathrm{Sq}}^{3}u$ is $u^2$.

The ring is exterior on its degree-$n$ universal class for odd $n$, and polynomial for even $n$. The path-loop Serre spectral sequence alternates these two types by transgression. In the polynomial-fiber case its derivation has nonzero coefficients $k$ on powers, which become invertible over the rationals.

For a CW complex $X$, abelian $G$ and $n\ge 1$, homotopy classes of maps to the Eilenberg–MacLane space correspond to cohomology classes by pullback of the universal cohomology class of an Eilenberg–MacLane space. Pointed maps give the reduced version. This converts a cohomology class into an actual map realizing its effect on homotopy.

For abelian $G$ and $n\ge 1$, the identity of $G$ determines this class under $H^n(K(G,n);G)\cong \mathrm{Hom}(G,G)$. Pulling it back realizes representability of cohomology by Eilenberg–MacLane spaces. A map inducing this class on a fiber induces the identity of its unique positive homotopy group.

A path-connected space is aspherical when its higher homotopy groups vanish, equivalently when its universal cover is contractible. An aspherical space with fundamental group $G$ is a $K(G,1)$, so based homotopy classes of maps into it are controlled by homomorphisms of fundamental groups.

Moving the basepoint around a loop acts on a higher homotopy group. Since ${\pi }_n(X)$ is an abelian group for $n\ge 2$, this action extends linearly to its integral group ring, making ${\pi }_n(X)$ a module. For a connected CW complex, its universal cover gives an equivalent description by deck transformations, together with change of basepoint upstairs. Left and right conventions are converted by inversion of the loop. The Hurewicz theorem respects this action whenever it identifies a higher homotopy group with the corresponding homology of the cover.

If $X$ is an $(n-1)$-connected based CW complex, $n\ge 2$, suspension induces an isomorphism ${\pi }_k(X)\to {\pi }_{k+1}(\Sigma X)$ for $k\le 2n-2$ and a surjection for $k=2n-1$. For a sphere, $\Sigma S^n\cong S^{n+1}$, giving the stable range ${\pi }_k(S^n)\cong {\pi }_{k+1}(S^{n+1})$.

Write the suspension of an $(n-2)$-connected based CW complex as two cones meeting in the original space. Each cone pair is $(n-1)$-connected because its relative groups shift the original homotopy groups by one. Homotopy excision theorem compares the cone pair with the suspension relative to the other cone, giving the suspension homomorphism as an isomorphism for $i<2n-2$ and a surjection at $i=2n-2$ in the indexing ${\pi }_{i-1}X\to {\pi }_i\Sigma X$.

The integral homology of the loop space of a sphere implies that the unit $S^{r-1}\to \Omega S^r$ has vanishing relative homology through degree $2r-3$. Both spaces are simply connected for $r\ge 3$. The absolute Hurewicz theorem first identifies their degree-two homotopy groups with homology; the relative Hurewicz theorem then rules out a first nonzero relative homotopy group in degrees three through $2r-3$. The relative homotopy sequence makes the unit an isomorphism on ${\pi }_i$ through degree $2r-4$. Loop-suspension adjunction identifies this map with suspension, yielding ${\pi }_i(S^{r-1})\cong {\pi }_{i+1}(S^r)$ in that range.

For $\varphi :S^{2q-1}\to S^q$, $q>1$, its mapping cone has integral cohomology generators $a$ in degree $q$ and $b$ in degree $2q$. The integer defined by $a⌣a=H(\varphi )b$, with fixed cell orientations, is the Hopf invariant. It measures how an attaching map changes multiplication without changing the additive groups. The complex Hopf fibration has invariant one because its mapping cone is ${\mathbb{CP}}^2$; a constant attaching map has invariant zero.

For the Hopf map $\eta$, the indicated four-cell attachment has cellular boundary multiplication by $f$ from dimension four to three. Its integral cohomology is $\mathbb{Z}$ in degrees zero and two, $\ker (f:\mathbb{Z}\to \mathbb{Z})$ in degree three, and $\mathbb{Z}/f\mathbb{Z}$ in degree four. If $x$ generates degree two and $z$ is the four-cell cochain class, then $x^2=dz$, with every other positive-degree product zero. Collapsing the three-sphere reduces the cup-square calculation to the Hopf invariant; the induced map on degree-four cohomology reduces its integer coefficient modulo $f$. When $f=0$ there is an extra free degree-three class, whose products still vanish by dimension.

Map $X_{d,f}$ to $K(\mathbb{Z},2)={\mathbb{CP}}^{\infty }$ using its degree-two generator. The homotopy fibre is the total space of the corresponding circle bundle, is 2-connected, and has ${\pi }_3$ equal to that of $X_{d,f}$. Over $S^2\vee S^3$ its total space is $P=S^3{\cup }_{S^1}(S^1\times S^3)$, with $H_3(P)=\mathbb{Z}A\oplus \mathbb{Z}B$. The four-cell bundle is trivial; its relative degree-four generator has boundary $dA+fB$, the lifted attaching map. The long exact sequence in relative homology and Hurewicz theorem therefore give ${\pi }_3(X_{d,f})={\mathbb{Z}}^2/⟨(d,f)⟩$. Smith normal form gives the displayed expression when $d>0$, including $f=0$.

Let $\varphi :S^{2q-1}\to S^q$ and precompose with a sphere self-map of topological degree $d$. The resulting map of mapping cones is the identity on the bottom $q$-cell and has degree $d$ on the top $2q$-cell. Pulling back the defining cup product relation for the Hopf invariant multiplies its coefficient by $d$. In particular the element $d\eta \in {\pi }_3(S^2)$ has Hopf invariant $d$. Postcomposition by a degree-$d$ map of the target sphere instead multiplies the invariant by $d^2$.

A map is a weak homotopy equivalence when it induces a bijection on path components and isomorphisms on every based homotopy group.

A map is $n$-connected when it induces isomorphisms on homotopy groups below degree $n$ and a surjection in degree $n$. Equivalently, each of its homotopy fibers is $(n-1)$-connected.

For a based space $X$, adjunction gives

The rational homotopy group is obtained by tensoring a homotopy group with the rational numbers. It retains the free part and discards torsion.

The Sullivan model of the connected sum of two complex projective planes has two generators in each of degrees two and three, and none elsewhere. Its rational homotopy groups are therefore exactly the displayed two copies of ${\mathbb{Q}}^2$, with all other positive degrees zero.

For an odd-dimensional sphere $S^{2m+1}$, only ${\pi }_{2m+1}(S^{2m+1})\otimes \mathbb{Q}$ is nonzero. For an even-dimensional sphere $S^{2m}$, the nonzero rational homotopy groups occur in degrees $2m$ and $4m-1$, and both are isomorphic to $\mathbb{Q}$.

If a path-connected space is $(n-1)$-connected for $n\ge 2$, then its first potentially nonzero homotopy group maps isomorphically to homology:

A sphere map $f:S^n\to X$ is sent to $f_{\ast }[S^n]\in H_n(X;\mathbb{Z})$. After tensoring with $\mathbb{Q}$ this gives the rational Hurewicz homomorphism. It is natural in $X$, sends Whitehead products to zero, and is an isomorphism in the first nonzero homotopy degree of a simply connected space by the Hurewicz theorem.

If $X$ is simply connected and $h_{\ast }:{\pi }_{\ast }(X)\otimes \mathbb{Q}\to {\tilde{H}}_{\ast }(X;\mathbb{Q})$ is onto, choose a homogeneous homology basis and representing sphere maps into $X_{\mathbb{Q}}$. Their wedge sum gives a rational homology equivalence, hence a rational homotopy equivalence by the rational Whitehead theorem. All positive cup products therefore vanish. In a minimal model, the induced map from cohomology to the indecomposable quotient is injective; replacing a set of generators by cocycle representatives produces a quasi-isomorphism to the square-zero cohomology algebra, proving formality of a topological space.

If $X$ and $A$ are simply connected and the pair is $(r-1)$-connected, with $r\ge 3$, its first possible nonzero relative homotopy group maps isomorphically to relative integral homology. This lets a basis of a free relative homology group be realized by attaching disks.

For a simply connected space, if lower homotopy groups lie in an admissible Serre class, the Hurewicz map in the next degree is an isomorphism modulo that class, and the lower homology groups lie in the class. This applies to finitely generated groups and to finite groups supported at prescribed primes.

A weak homotopy equivalence between connected CW complexes is a homotopy equivalence. The analogous statement holds for Kan complexes.

For maps between simply connected spaces, a rational homology equivalence is a rational homotopy equivalence. It follows from the relative Hurewicz theorem modulo torsion, or from rationalization and the homological Whitehead theorem.

An integral homology equivalence between simply connected spaces is a weak homotopy equivalence. A first nonzero relative homotopy group would contradict the Relative Hurewicz theorem. Between CW complexes the ordinary Whitehead theorem then gives a homotopy equivalence.