nLab James construction

Idea

The free topological monoid $F X$ on a topological space $X$ is canonically filtered by the length of words. Given instead a pointed topological space $(X,x)$, there is also a reduced version by taking $F X$ and identifying $x$ with the identity of $F X$. This latter filtered topological space is known as the James construction $J(X,x)$ (James 55).

Details

The James construction $J(X,x)$ may be constructed homotopy-theoretically (Brunerie 13, Brunerie 17). Recall that, for $K$ a finite simplicial complex, for $(X,A)$ a pair of spaces, its polyhedral product $(X,A)^K$ is defined as the union $\bigcup_{\sigma\in S(K)}(X,A)^\sigma$ as a subspace of the Cartesian product $X^{V(K)}$. Here, for $\sigma\in S(K)$ a simplex of $K$, the subspace $(X,A)^\sigma$ consist of those $x\in X^{V(K)}$ such that, for each vertex $v$ in the complement of $\sigma$, the coordinate projection $\proj_v x$ lies in $A$. Equivalently, the polyhedral product $(X,A)^K$ can be considered as a homotopy colimit of these $(X,A)^\sigma$ over the poset $S(K)$ of simplexes $\sigma$ of $K$, where the maps are the respective inclusions.

Definition

For $X$ a space equipped with a basepoint $x$, define a filtered space $fil_\bullet$ as follows. Set $\fil_0$ as $\{x\}$. For $k\ge 1$, require that the following square is homotopy pushout:

$$\array{ &&&& (X,x)^{\partial \Delta[k-1]} &&&& \\ & && inc \swarrow & & \searrow && \\ && X^k &&&& fil_{k-1} \\ & && {}_{p_k}\searrow & & \swarrow_{j_k} && \\ &&&& fil_k &&&& }$$

where the unlabeled arrow is the homotopy colimit of a morphism of diagrams over $S(\partial \Delta[k-1])$ given by the maps

$$(X,x)^\sigma \xrightarrow{\sim} X^{\dim(\sigma)+1} \xrightarrow{p_{\dim(\sigma)+1}} fil_{\dim(\sigma)+1}$$

for each simplex $\sigma$ of the boundary simplicial complex $\partial \Delta[k-1]$ of the standard $(k-1)$-simplex. The homotopy colimit $fil_\infty$ of the sequence of maps $fil_0 \stackrel{j_1}{\to} fil_1 \stackrel{j_2}{\to} \ldots$ is called the James construction on $(X, x)$.

Proposition

For $(X,x)$ a pointed space, if $(X,x)$ is path-connected, then $fil_\infty \simeq \Omega\Sigma X$.

Properties

Relation to configuration spaces

The James construction of $X$ is homotopy equivalent to the configuration space $C(\mathbb{R}^1, X)$ of points in the real line with “charges” taking values in $X$.

(e.g. Bödigheimer 87, Example 9)

See also

• James construction type

References

The construction is due to

• Ioan James, Reduced product spaces, Annals of Mathematics, Second Series, 62: 170–197 (1955)

Review:

• Michael Hopkins (notes by Akhil Mathew), Lectures 2,3 in: Spectra and stable homotopy theory, 2012 (pdf, pdf)

• Dylan Wilson, James construction, 2017 (pdf)

• Wikipedia, James reduced product

Discussion via configuration spaces includes

• Carl-Friedrich Bödigheimer, Stable splittings of mapping spaces, Algebraic topology. Springer 1987. 174-187 (pdf)

Discussion via homotopy type theory includes the following

• Guillaume BrunerieThe James Construction and $\pi_4(S^3)$, talk at the Institute of Advanced Studies on March 27, 2013 (recording)

• Guillaume Brunerie, The James construction and $\pi_4(S^3)$ in homotopy type theory (arXiv:1710.10307)