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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.

Because $S^n$ is simply connected for $n\ge 2$, the homological Serre spectral sequence has constant coefficients and only two nonzero columns:Its only possible nonzero differential isUsing the orientation generator of $H_n(S^n)$ to identify both columns with $H_{\ast }(F)$ defines the Wang homomorphismThe kernel and cokernel descriptions of the two surviving columns splice with the filtration of $H_{\ast }(E)$ to give the Wang sequence over a sphere

Let $F_m$ be the homotopy fiber of a degree-$m$ map $f_m:S^n\to S^n$. Apply this sequence to the fibrationAt the bottom, the map between the two copies of $H_n(S^n)$ is multiplication by $m$. It follows thatThe first torsion group can also be seen from ${\pi }_{n-1}(F_m)\cong \mathbb{Z}/m$ and the Hurewicz theorem; the Wang map then propagates it periodically.

The universal coefficient theorem for cohomology givesEvery product of two positive-degree classes is zero. For $n>2$, this follows immediately because the sum of two degrees of the form $k(n-1)+1$ is not of that form. For $n=2$, degree counting does not suffice, since $H^j(F_m)\cong \mathbb{Z}/m$ for every $j\ge 2$. Use instead the multiplicative cohomological Serre spectral sequence forIts transgression in degree one is multiplication by $m$. Every positive integral cohomology class that survives lies in filtration two, while the product of two such classes lies in filtration four; the base has dimension two, so filtration four is zero. Thus the reduced cohomology is a square-zero ideal, andas a graded ring, with zero multiplication on the second summand.