Unified L-theory

An object x is closed if Ox = 0, and it is a boundary if x = 3y for some object y. In particular, boundary objects are closed, and if x is closed (resp. a boundary) then so is -x. For n 0 the closed objects are precisely the algebraic Poincare complexes, and for n 1 they are precisely the non-singular forms and formations. 

Given objects x,y in the same i-category and a homotopy equivalence of the boundaries of x and -y f 3x -3y 

A cobordism (z f,g) of objects x,y in the same n-dimensional -category is a triple consisting of an object z of the corresponding (n+1) dimensional /.-category, and homotopy equivalences 

For example, if (M,iJ)) is an e-symmetric form over A and LCm is a sublagrangian the operation 

Note that 3x = i x in each case, so that the homotopy type 

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In SI.8 below we shall recall from SI.6 the way in which the even c-symmetric L-groups L (A,c) for n - 0,1 bridge the gap between the c-quadratic and the e-symmetric L-groups, defining a unified L-theory containing all three types of L-gr< Proposition 1.2.2 iii) also extends to the even E-symmeti L-groups, with L (A,c) O for a 0-diinensional ting with involution A (cf. the proof of Proposition 1.4.51. 

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