## A composition formula for manifold structures.(English)Zbl 1193.57011

Let $$M$$ be an $$n$$-dimensional topological manifold. The structure set $$\mathcal{S}^{TOP} (M)$$ is the basic object for the classification of manifolds which are homotopy equivalent to $$M$$. Strictly speaking, the set $$\mathcal{S}^{TOP} (M)$$ is the pointed set of equivalence classes of pairs $$(N,f)$$ with $$N$$ an $$n$$-dimensional manifold and $$f: N\to M$$ a homotopy equivalence, with $$(N_1,f_1)=(N_2,f_2)\in \mathcal{S}^{TOP} (M)$$ if and only if $$(f_1)^{-1}f_2: N_2\to N_1$$ is homotopic to a homeomorphism, and $$(M,1)\in \mathcal{S}^{TOP} (M)$$ the base point. For $$n\geq 5$$, the structure set $$\mathcal{S}^{TOP} (M)$$ fits into the topological Browder-Novikov-Sullivan-Wall surgery exact sequence, which is isomorphic to the algebraic surgery exact sequence which was constructed by the author in [Algebraic $$L$$-theory and topological manifolds, Cambridge Tracts in Mathematics. 102. Cambridge: Cambridge University Press (1992; Zbl 0767.57002)]. This isomorhism provides a natural abelian group structure with operation $$+$$ on the structure set $$\mathcal{S}^{TOP} (M)$$.
Let $$g: P\to N$$, $$f: N\to M$$ be homoptopy equivalences with $$(N,f)\in \mathcal{S}^{TOP} (M)$$, $$(P,g)\in \mathcal{S}^{TOP} (N)$$. In the paper under review, the author proves the composition formula $$(P,fg)=(N,f)+f_*(P,g)\in \mathcal{S}^{TOP} (M)$$, where $$f_*$$ is a functorial map of algebraic surgery exact sequences induced by $$f$$. The author also describes a relation of the obtained formulae to two natural abelian group structures on the set of normal bordism maps to the manifold $$M$$. Then the author obtains results about a composition of topological normal maps and about the Whitney sum of topological normal invariants. The paper is concluded with an example of application of the results to $$\mathcal{S}^{TOP} (S^p\times S^q)$$.

### MSC:

 57N70 Cobordism and concordance in topological manifolds 57N65 Algebraic topology of manifolds 19J25 Surgery obstructions ($$K$$-theoretic aspects) 19G24 $$L$$-theory of group rings 57P10 Poincaré duality spaces 57R67 Surgery obstructions, Wall groups

Zbl 0767.57002
Full Text: