Loop space ($\Omega X$)

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 $X$.

For based continuous maps $\alpha :S^p\to G$ and $\beta :S^q\to G$ into a topological group, take the pointwise commutatorIt is exactly the identity on $S^p\vee S^q\subseteq S^p\times S^q$, so it descends to the smash product $S^p\wedge S^q\cong S^{p+q}$. Its homotopy class is the Samelson product $⟨\alpha ,\beta ⟩\in {\pi }_{p+q}(G)$. This measures failure of the two maps to commute; it vanishes for an abelian group.^[ref]

For an ordinary loop space, reversal is an inverse up to homotopy, and a retraced loop need not be the constant loop. Literal descent therefore requires a strict topological group model or the corresponding coherent contractions, rather than treating loop reversal as a strict inverse. Equivalently, suspension-loop adjunction defines the operation through the sign-adjusted Whitehead product in the original space.

In positive degrees these operations give the graded antisymmetry and Jacobi identity of the loop-space homotopy Lie algebra. Under the Hurewicz homomorphism, the product becomes the graded commutator in the Pontryagin ring.

The based path space $P_{x_0}X$ consists of continuous paths beginning at $x_0$, with variable endpoint and the compact-open topology. Endpoint evaluation is a Serre fibration with fiber the based loop space. Its total space is contractible by $\gamma (t)\mapsto \gamma (st)$ as $s$ decreases to zero. This differs from a space of paths with both endpoints fixed, which need not be contractible. The long exact sequence of homotopy groups of a fibration gives ${\pi }_i(\Omega X)\cong {\pi }_{i+1}(X)$ for connected $X$ in the usual based ranges.

For $r\ge 3$, the path-loop Serre fibration has contractible total space. The relative homology of a fibration over a sphere gives $H_j(\Omega S^r)\cong H_{j-r+1}(\Omega S^r)$ for $j>0$, with negative homology zero and $H_0=\mathbb{Z}$. This proves the displayed groups. The unit map $S^{r-1}\to \Omega S^r$, adjoint to the identity of $S^r$, represents the generator of the first nonzero homotopy group. Naturality of the Hurewicz theorem shows that it induces an integral homology isomorphism in all degrees through $2r-3$.

For $n\ge 1$, there is one integral cohomology generator $a_k$ in every degree $2nk$, and its products are $a_i{a}_j=\left(\genfrac{}{}{0ex}{}{i+j}{i}\right)a_{i+j}$. The Bott–Samelson theorem supplies free polynomial homology; its primitive generator gives the binomial diagonal dual to this divided power algebra.

The homology ring formed from loop concatenation and the homology cross product. Homotopy associativity is enough for associativity in homology, and the constant loop is the unit. More generally the same construction applies to a homotopy-associative H-space.

For simply connected $Y$, rational Hurewicz homomorphism identifies ${\pi }_{\ast }(\Omega Y)\otimes \mathbb{Q}$ with the primitive elements of a Hopf algebra of $H_{\ast }(\Omega Y;\mathbb{Q})$ and induces an isomorphism $U({\pi }_{\ast }(\Omega Y)\otimes \mathbb{Q})\cong H_{\ast }(\Omega Y;\mathbb{Q})$ of graded Hopf algebras. The bracket on homotopy is the Samelson product. The underlying algebra theorem says that a connected graded cocommutative Hopf algebra over characteristic zero is the universal enveloping algebra of its primitive Lie algebra. These statements let a free tensor algebra of loop-space homology detect a free graded Lie algebra of homotopy.

For $n\ge 1$, the Bott–Samelson theorem gives a single homology generator of degree $2n$ with no word relations. Its Pontryagin product satisfies $x^i\ast x^j=x^{i+j}$. Unlike this polynomial homology product, the cohomology cup product has divided-power coefficients.

Apply the homology cross product and then the map induced by loop concatenation. This degree-additive bilinear product supplies the Pontryagin ring. The Bott–Samelson theorem computes it for looped suspensions with free homology.

Finite words in a based space, with occurrences of its basepoint deleted. Word concatenation defines a multiplication. For connected based CW spaces, the natural map to $\Omega \Sigma X$ is a weak homotopy equivalence. With free integral homology, the Bott–Samelson theorem identifies its homology algebra with the tensor algebra on reduced homology.

For a connected based CW complex with free integral homology, the Pontryagin ring of its looped reduced suspension is the indicated tensor algebra. The generator inclusion comes from $X\to \Omega \Sigma X$, and multiplication corresponds to concatenation of words in the James reduced product.

Loop-space homology is the homology of the based loop space. It also carries the Pontryagin product induced by concatenation of loops. For $n>2$, a Morse theory cell structure on $\Omega S^n$ has one cell in each dimension $j(n-1)$, so its integer homology is $\mathbb{Z}$ in those dimensions and zero elsewhere.

The evaluation map from the based path space gives a fiber sequenceSince $PX$ is contractible, it relates invariants of $X$ to those of its loop space.

Pull back along loop evaluation $S^1\times \Omega B\to B$ and take the slant product with the circle's fundamental class. This degree-lowering operation kills positive-degree cup products. It commutes with stable cohomology operations such as Steenrod squares and identifies a transgressing base class with its fiber class in the path-loop fibration.