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If a continuous retraction from a closed disc to its boundary existed, applying it to a contraction of the boundary circle inside the disc would give a homotopy in $\mathbb{C}\setminus \{0\}$ from a loop of winding number one to a constant loop of winding number zero. Homotopy invariance of winding number rules this out.

For a polynomial $P(z)=a_n{z}^n+\cdots +a_0$ of positive degree, on a sufficiently large circle its leading term dominates the remaining terms. The dominated-perturbation lemma therefore gives winding number $n$ to the loop $P(R{e}^{2\pi it})$. If $P$ had no zero, radial contraction of the input circle would map under $P$ to a homotopy with a constant loop in $\mathbb{C}\setminus \{0\}$, which has winding number zero. This contradiction proves the Fundamental theorem of algebra.

If closed continuous paths $\gamma ,\varphi$ satisfy $|\gamma (t)|>|\varphi (t)|$ for every $t$, then $H(s,t)=\gamma (t)+s\varphi (t)$ never vanishes. It is therefore a homotopy through closed paths in $\mathbb{C}\setminus \{0\}$, and

The Steenrod squares are natural operationsFor a degree-one class $x$, ${\mathrm{Sq}}^{1}x=x^2$, and ${\mathrm{Sq}}^1$ agrees with the mod-two Bockstein.

For a fibration $F\to E\to B$ with suitable connectivity, the cohomological Serre spectral sequence haswith differential $d_r:E_r^{p,q}\to E_r^{p+r,q-r+1}$.

Simplicial homotopy theory models spaces and higher categories using simplicial sets.

The loop space $\Omega X$ is the space of based loops in $X$. It is the homotopy fiber of the inclusion of the base point into $X$.

A homotopy pullback is the derived form of a pullback. A strict pullback square of fibrant objects computes a homotopy pullback whenever one of the two maps into the lower-right object is a fibration.

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.

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 $n\ge 2$.

A pair $(X,A)$ has the homotopy extension property when every homotopy on $A$ whose initial map extends to $X$ can itself be extended to a homotopy on $X$.

A subspace $A\subseteq X$ is a retract of $X$ when there is a continuous map $r:X\to A$ whose restriction to $A$ is the identity.

Spaces $X$ and $Y$ are homotopy equivalent when there are continuous mapssuch that $g\circ f$ is homotopic to ${\mathrm{id}}_X$ and $f\circ g$ is homotopic to ${\mathrm{id}}_Y$.

A space is contractible when its identity map is homotopic to a constant map. Every map into a contractible space is null-homotopic.

A map is null-homotopic when it is homotopic to a constant map.

The local homology at $x\in X$ is $H_i(X,X\setminus \{x\})$. It is preserved by homeomorphisms and, in a simplicial complex, is computed from the reduced homology of the local link.

Simplicial homology computes homology from the boundary maps between free abelian groups generated by oriented simplices.

For an abelian coefficient group $G$, there is a split short exact sequenceFor $G=\mathbb{Q}$, the Tor term vanishes.

This is the homology case of the universal coefficient theorem; its cohomology case uses Hom and Ext rather than tensor product and Tor.

A chain complex is a sequence of abelian groups or modules and homomorphisms $d_i:C_i\to C_{i-1}$ satisfying $d_{i-1}{d}_i=0$. Its homology is $H_i(C)=\ker d_i/\mathrm{im}{d}_{i+1}$.

A commutative diagram is a diagram of objects and morphisms in which every two directed paths with the same endpoints define the same morphism.