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Construction of isology functors

Arthur H. Copeland

1966
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Transactions of the American Mathematical Society
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This paper is, hopefully, a step in the direction of obtaining a complete system of algebraic invariants for the isotopy classes of finitely triangulable spaces. We construct a large family of isotopy functors defined on a category of embeddings of topological pairs and having values in the category of homomorphisms of graded abelian groups. These functors are formally like homotopy and homology functors, having a boundary operator and satisfying an exactness axiom, and some of them are
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... of them are especially well behaved: they are isology functors, which is to say, they satisfy an excision axiom. These functors are defined in the first three sections and their basic properties are established. The rest of the paper is concerned with testing whether these functors can tell one space from another. Ideally, one should construct a functor Lwith the property that iff is an embedding and if T(f) is an isomorphism, then/is an isotopy equivalence. In fact, somewhat less than this has happened. The ideal situation holds for embeddings of 1-dimensional finitely triangulable spaces, but at the next level of complexity, embeddings of 2-dimensional finitely triangulable cone spaces, there are nonequivalences which induce isomorphisms. This gap may be partially closed by employing the deleted product functor, but there still remain some spaces which look alike to all of the functors that the author knows how to deal with gracefully. However, it should be noted that the remaining cone spaces are ' quite simple and it seems unlikely that they will resist algebraic analysis for long. 1. The basic functors. If X is a topological space, let cX= X*p be the join of X with a point p = aX not in X. The pair (cX, aX) is called a cone over X, cX is the space of the cone and aX is the apex. The space X is naturally embedded in cX and is called the base of the cone. A map/: X ->-F induces a map cf: cX, aX -> cY, a Y such that each segment from xelto aX is mapped linearly onto the segment from f(x) to a Y. Any two cones over the same space are topologically equivalent pairs, and we choose to ignore the difference between them. In this spirit, we demand that if Y<= X, then aY=aX and cY<=cX, so that c is now regarded as a covariant functor from maps between topological pairs to maps between pairs with basepoint. If / is an embedding, then cf is also an embedding, and if / is isotopic to g, then cf is isotopic to eg; the second isotopy leaves the apex fixed. If X and K are topological spaces, let m(X; K) be the following subspace of K, m(X; K) = {/(aAOI/embeds cXin K}.

doi:10.1090/s0002-9947-1966-0203735-3
fatcat:p2rinrw5cvgixk663rogd5yf5a