Tim Porter Strong shape, coherent prohomotopy, and the stability problem

Strong shape, coherent prohomotopy, and the stability problem:

Shape theory

In the work for his DPhil thesis (1968-71), Porter developed a form of Čech homotopy bearing the same relationship to the usual homotopy as Čech homology had to ordinary singular homology. This turned out to be equivalent to Borsuk’s theory of Shape, for compact metric spaces, and to the Mardesic-Segal version for compact Hausdorff spaces.

The stability problem

One central problem identified in this work, as a focus for future development, was the stability problem. In this, the point is not only to find criteria that will imply that a space has the same shape as a polyhedron, but also to try to measure the deviation of ‘non-stable space from stability, that is, how far away it was from being ‘stable’. (Later this was linked to the theory of stable ends in the sense of Siebenmann.) This was related to a problem, the movablity problem, that Borsuk had identified at the start of his work on Shape Theory.

Porter’s answer to this problem, in his paper of 1974, was innovative. It used the Vietoris complex construction to get a pro-simplicial set from a space $X$, then methods of homotopy limits as, then recently, developed by Vogt, Bousfield and Kan, to obtain a candidate CW-complex, $V(X)$, so if $X$ was stable, it would stabilise to this $V(X)$, and if not then there was a possibility of discovering invariants that would measure its deviation from stability by comparing $X$ with $V(X)$, for instance by localising at some prime, under suitable conditions. Certain criteria for stability were developed, which gave a partial answer. (The problem was solved a short time afterwards by Edwards and Geoghegan, and separately by Morita using related methods.)

Coherent prohomotopy and strong shape

Abstract homotopy theories on a procategory

The importance of this approach to the stability problem was that it emphasised a need for some abstract homotopy structure on the category, $Pro-S$, of pro-simplicial sets. This had been already noted by Artin and Mazur in their lecture notes on étale homotopy. Without such a structure available, they had worked with $Pro-Ho(S)$ rather than $Ho(Pro-S)$, but realised the advantages of the latter if such a structure could be found, and if the hypercovering methods that they had used could be made to be to lead to objects in that better structured setting. The problem of abstract homotopy theories on pro-categories is a hard one due to the very complex nature of the morphisms involved. In fact, it is only in the last few years that the proposed solutions seems to have reached a final form with work by Isaksen.

Coherent prohomotopy and obstruction theory

Porter’s attack on this problem was motivated by less abstract problems, firstly that of the stability problem mention above, (where a version of $Ho(Pro-S)$ was introduced and used), and then by the problem of developing an obstruction theory for shape theory and, hopefully, étale homotopy. In fact these problems related more to strong shape theory, that is one based on $Ho(Pro-S)$ rather than the weak form, and the use of $V(X)$ as mentioned above.

Porter noted that it was sufficient to use K.S. Brown’s (co)fibration category structure as a basis for the obstruction theory, rather than a full blooded Quillen model category structure. Using this he developed an abstract homotopy theory for $Pro-S$, a related homological algebra for $Pro-Ch(Ab)$, the category of pro-chain complexes of Abelian groups, and then combining them to give an obstruction theory applicable in $Pro-S$. This was then applied to problems in strong shape theory and a ‘coherent’ form of étale homotopy. This well developed homotopy theory was a coherent prohomotopy theory, which meant that both objects and morphisms had an interpretation as coming from homotopy coherent diagrams. (A very closely related Quillen model category structure was published at about the same time by Edwards and Hastings and was applied by them to proper homotopy theory as well as what they termed Steenrod homotopy theory. Their theory gave an equivalent category $Ho(Pro-S)$. This forms the main part of Isaksen’s recently proposed prohomotopy structure.)

Proper homotopy theory

Another offshoot of this work was the work by Porter with Hernandez on proper homotopy theory. This came from a combination of the above with the Edwards-Hastings theory of proper homotopy, as being a dual form of their strong shape / Steenrod homotopy theory. This resulted in an invitation to write a chapter for the Handbook of Algebraic Topology.