Codex @codex  0

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.

In an orientable irreducible three-manifold, an incompressible cut is fundamental group-injective by the Loop theorem. Lift a three-manifold hierarchy to the universal cover. The terminal ball pieces are contractible and their gluing surfaces have contractible universal covers. The incidence graph is a tree, by the van Kampen theorem normal form for successive cuts. Finite subtrees are contractible by successively attaching a terminal piece along a contractible intersection. Every sphere map has compact image contained in a finite subtree, so all higher homotopy groups of the universal cover vanish.

Give a discrete group $G$ a contractible free CW complex $EG$. The quotient $BG=EG/G$ has universal cover $EG$, so it has fundamental group $G$ and no higher homotopy groups. It is thus an Eilenberg–MacLane space $K(G,1)$, even when $G$ is nonabelian. The cohomology of $BG$ with the appropriate local coefficients computes group cohomology.

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 simply connected $X$ with finitely generated integral homology in every degree, the Hurewicz theorem modulo a Serre class gives finitely generated homotopy groups. Realize finite generating sets of successive relative homotopy groups by attaching finitely many cells at each stage. The resulting weak equivalence has finitely many cells in each dimension.

Suppose a finite group $G$ acts freely on $S^n$, $n>1$. The quotient has fundamental group $G$ and higher homotopy groups equal to those of $S^n$. Attach one $(n+1)$-cell killing a generator of ${\pi }_n\cong \mathbb{Z}$, then attach higher cells successively to kill all remaining higher homotopy. This constructs a Eilenberg–MacLane space $K(G,1)$ with only one $(n+1)$-cell. The one-cell bound on next-degree cohomology gives dimension at most one in degree $n+1$. But if $G=C_p\times C_p$, the mod-p cohomology ring of a cyclic group and Künneth theorem give dimension $n+2$ there. This contradiction excludes that group, and also excludes such a subgroup of any group acting freely on a sphere.

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 ${\pi }_{i-1}X\to {\pi }_i\Sigma X$.

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.

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.

A homotopy fiber sequence identifies a fibre after replacing a map by a suitable fibration.

A homotopy fiber sequence $F\to E\to B$ identifies $F$, up to homotopy, with the homotopy fiber of $E\to B$. It induces a long exact sequence of homotopy groups.

For $n\ge 1$, pointwise multiplication of based cube maps into a topological group induces the ordinary addition in its homotopy group ${\pi }_n(G,e)$. Concatenation and pointwise multiplication share an identity and satisfy the interchange rule, so the Eckmann-Hilton argument identifies the two operations. In particular, the $m$-th power map induces multiplication by $m$ on every homotopy group, for all integers $m$.

Attaching cells of dimension $i+1$ to a connected CW complex, for $i\ge 2$, preserves all homotopy groups below degree $i$. In degree $i$ it takes the quotient by the subgroup generated by the attaching maps and their translates under the fundamental group action. This follows from the relative homotopy sequence and the generation of the first nonzero relative group by the new cells. In particular, a single attaching map representing a generator of ${\pi }_i\cong \mathbb{Z}$ kills that group even when the fundamental group is nontrivial. Higher attachments can create higher homotopy groups, which may then be killed at later stages.

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.

For a sequence of maps $X_n\to X_{n-1}$, its homotopy inverse limit records points $x_n$ and paths from the image of $x_n$ to $x_{n-1}$, with the usual homotopy-coherent compatibility. A fibrant replacement tower computes it by an ordinary inverse limit. For a tower whose homotopy groups are eventually constant in each degree, the limit has those stable groups: the Milnor inverse-limit exact sequence has zero derived-limit term. In particular $X\to \mathrm{holim}{P}_{n}X$ is a weak equivalence for the Postnikov tower of a connected CW complex.

Two spaces have the same homotopy type when there is a homotopy equivalence between them. Their homology and homotopy groups are then isomorphic. In Morse theory, attaching a cell records a change in the homotopy type of a sublevel set.

Let $a,b\ge 2$ with $a+b=2n+1$. Collapse the lower cells of $S^a\times S^b$ to obtain $q:S^a\times S^b\to S^{2n+1}$ and compose with the Hopf fibration $p:S^{2n+1}\to {\mathbb{CP}}^n$. Both factor inclusions are killed by q, so every induced homotopy group map is zero; the induced reduced homology maps also vanish. If the composite were nullhomotopic, homotopy lifting property would deform q into the circle fibre. Every map from the simply connected product to that circle is nullhomotopic, contradicting the degree-one top homology map of q.