# 4-Manifolds which embed in R5 R6, and Seifert manifolds for by Cochran T.

By Cochran T.

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Extra info for 4-Manifolds which embed in R5 R6, and Seifert manifolds for fibered knots

Example text

Choose an alter ego M of M and define X :-: IScP^(M). We have seen that, for each A e A, there is a natural embedding e^ : A ^-^ ED (A) given by epj^a)[x) :^ x(a), for all a G A and x G A{A, M). Similarly, for each X G X, there is a natural embedding 6x : X ^ DE(X), given by evaluation: ex(3^)(<^) '= a{x), for all x G X and a G X(X, M). We say that M yields a full duality on A (based on M) if the maps e^ and ^x are isomorphisms, for all A G ^l and X G X. In this case, the categories A and X are dually equivalent.

For any a ^ A, the map a agrees with CjsJ^a) on a~^{s) if and only if, for all x G a~^{s), we have x{a) = a{x) — s. So a agrees with an evaluation on a~^ (s) if and only if the set Aa,s :=f]{x-\s) I xea-\s)} is non-empty. We are aiming to find a dualising structure for M that is of finite type. 2, we can assume that the algebra A is finite. Now the set A^^s is non-empty provided that (a) the set x~^{s) is non-empty, for each x G a~^{s), and (b) the set Pdcx^s '-= { x'~^{s) \ x G a~^{s) } is closed under pairwise intersection.

Before stating the lemma, we present some more definitions. Consider an algebra A ^ A and a map a : yi(A, M) -^ M. For each a G A, we say that a is given by evaluation at a if a — epj^a). For Y C A{A, M) and a ^ A, we say that a is given by evaluation at a on y if afy — e^(<^) ly- The map a is said to be locally an evaluation if it agrees with an evaluation on every finite subset of yi(A, M). 6]. 5 Brute Force Lemma Let M be a finite algebra. Define A := I§P(M) and let A e A. Then a map a : A{A, M) —> M is a brute-force morphism if and only if a has a finite support and is locally an evaluation.