Homotopy theory

Homotopy theory studies topological spaces and continuous maps through continuous deformation. A homotopy from $f:X\to Y$ to $g:X\to Y$ is a continuous map $H:X\times [0,1]\to Y$ with $H(x,0)=f(x)$ and $H(x,1)=g(x)$. Replacing a space by a homotopy equivalent one preserves its deformation-level information, including its homotopy groups.

For example, the punctured plane retracts onto its unit circle throughThe coefficient stays positive, so the deformation never crosses the removed origin, and points already on the unit circle remain fixed. This is a deformation retraction.

The same viewpoint studies loops, higher-dimensional spheres and maps between spaces. It retains obstructions that survive continuous deformation while allowing distances, angles and the thickness of a region to change. A homotopy equivalence is weaker than a homeomorphism.

A standard form of the Sullivan conjecture, proved by Haynes Miller, states that for a finite group $G$ and a connected finite-dimensional CW complex $X$, the pointed mapping space from the classifying space of a discrete group $BG$ to $X$ is a weakly contractible space:Thus every basepoint-preserving continuous map $BG\to X$ is null-homotopic, and higher homotopy families of such maps are trivial as well. Miller's Theorem A allows the more general assumption that every finitely generated subgroup of $G$ is finite.^[ref]

The finite-dimensional target hypothesis is substantial. For $G=\mathbb{Z}/2\mathbb{Z}$, maps from $BG$ to a finite-dimensional sphere satisfy the theorem, while the identity map of $BG$ is not null-homotopic: it induces the identity on its nontrivial fundamental group. That target $BG$ is infinite-dimensional. The theorem concerns these unstable mapping spaces; it cannot simply be transferred to every stable mapping construction.

Chromatic homotopy theory organizes stable homotopy theory by a prime number $p$ and a hierarchy of heights $n$. Its main probes include Morava K-theory, whose standard coefficient ring for $n\ge 1$ isThe index is the height of the associated formal group law. Rational homology plays the height-zero role. These different probes detect different parts of a spectrum, rather than replacing all stable homotopy information by one ordinary homology group.^[ref]

A basic chromatic question asks which heights detect a given finite spectrum: its type is the least $n$ with $K(n{)}_{\ast }(X)\ne 0$, when such an $n$ exists. Rational information already detects the sphere spectrum at height zero, whereas purely torsion phenomena require other probes. The chromatic viewpoint makes periodic patterns and localization of spectra into organizing tools for difficult stable computations.

At a prime number $p$ and height $n\ge 1$, Morava K-theory is a periodic generalized homology theory represented by a topological spectrum $K(n)$ with coefficient graded ringHere multiplication by $v_n$ is invertible and implements periodicity. Every nonzero homogeneous coefficient is invertible, so this is a graded field, although it is not a field as an ungraded ring.^[ref]

For $p=2,n=1$, the period is two: the coefficient group is ${\mathbb{F}}_2$ in each even degree and zero in odd degrees. Distinct heights supply different tests for topological spectra; vanishing of $K(n{)}_{\ast }(X)$ means that this particular probe misses $X$, not that $X$ is necessarily trivial. This is the organizing idea of chromatic homotopy theory.