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Calculus. I: The first derivative of pseudoisotopy theory. (English) Zbl 0741.57021
A $$k$$-connected map $$Y\to X$$ of manifolds induces a $$(k-2)$$-connected map $${\mathcal P}(Y)\to {\mathcal P}(X)$$ of the associated stable smooth pseudoisotopy spaces. The main result of the paper describes the $$(2k- 3)$$-homotopy type of $$fiber({\mathcal P}(Y)\to {\mathcal P}(X))$$. Let $$\Lambda(X)=Maps(S^ 1,X)$$ denote the free loop space and $$\Lambda(Y\to X)=Y\times_ X\Lambda(X)$$. The inclusion of constant loops $$X\to\Lambda(X)$$ induces an inclusion $$Y\to \Lambda(Y\to X)$$. The $$(2k- 3)$$-homotopy type of $$fiber({\mathcal P}(Y)\to{\mathcal P}(X))$$ is the same as that one of $$fiber(\Omega^ 2Q(\Lambda(Y\to X)/Y)\to \Omega^ 2Q(\Lambda(X)/X))$$ with $$Q=\Omega^ \infty\Sigma^ \infty$$. As a corollary the author determines the $$(2k-3)$$-homotopy type of $$fiber(A(Y)\to A(X))$$, Waldhausen’s relative $$K$$-theory of topological spaces. Using the theory of calculus of functors the proof reduces to a geometric analysis of the map $$P(N)\to P(M)$$ of the unstable pseudoisotopy spaces induced by the inclusion $$N\subset M$$, where $$M$$ is obtained from $$N$$ by attaching a handle. The calculus of functors, developed by the author in the past decade, is a theory particularly suited for stable range calculations and has had remarkable applications. The paper starts with a concise treatment of the differentiation and the derivative of homotopy functors (higher derivatives and the powerful theory of analytic functors are deferred to a later paper). The derivative of $$X\mapsto Q(Map(K,X)_ +)$$, $$K$$ a finite complex, is determined in Section 2. This example is of special interest in view of the connection of $$Map(S^ 1,X)$$ to algebraic $$K$$-theory. The main result is phrased as calculation of the derivative of the functor $$X\mapsto {\mathcal P}(X)$$, which determines the $$(2k-3)$$-homotopy type of $$fiber({\mathcal P}(Y)\to {\mathcal P}(X))$$. Its proof constitutes Section 3.

##### MSC:
 57R52 Isotopy in differential topology 19D10 Algebraic $$K$$-theory of spaces 55P65 Homotopy functors in algebraic topology
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##### References:
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