By Francesco Costantino

We determine a calculus for branched spines of 3-manifolds through branched Matveev-Piergallini strikes and branched bubble-moves. We in brief point out a few of its attainable functions within the research and definition of State-Sum Quantum Invariants.

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R y-Z defined by S'(g)oa x R FZ. Then S' oS(f) = for = each S' (S(f)) = zrRX(g ) X f. e for Z, each g E S'(S(f))oa X On the o t h e r hand, YIRZ = and X rRX(S(f)) let g e 1TRZ. E )~. =, Let f o a X, Then X, lrRX(S(S' (g))) = S ' ( g ) o a x = rRX(g), so t h a t S o S ' (g) = S(S' (g)) = g. for f so E that each X E • Like the product function, the sum function can he n a t u r a l l y r e s t r i c t e d to continuous functions. If )~ is a family of spaces, let C(Z,R) or C ( f ) denote the family {C(X,R) : x~z}.

2. the that induced ¢(Ca(X)) function < ¢(Ca(Y)) ,, The following are equivalent, (a) Each point of Ck(X ) is a G 6 - s e t . (b) Each compact subset of Ck(X ) is a G 6 - s e t . (c) Ck(X ) has a (]6 diagonal. (d) Ck(X ) is submetrizable. (e) X is almost a - c o m p a c t . 3. p : Ca(X ) ~ The following are equivalent. = C a ( Y ) is a continuous ¢(II{Ca(A): A e ~9}) < 57 (a) Each point of Cp(X) is a G 3 - s e t . (b) Each compact subset of Cp(X) is a G 3 - s e t . (c) Cp(X) has a G$ diagonal. (d) Cp(X)is submetrizable.

Aanw(X). anw(X) _< aanw(X). For of X r. the with r e v e r s e inequality, let r ~ I~/| B be _< Define ~/MB = X, t h e n A c so that 1 ~/nB I if B c UMB is _< r . of a closed of X, let A E a V for some B E an a-network of X a c o n s i s t s of the ~2 c a-network a of be an X a-cover with B}, which is a s u b s e t of a. and B. Let tB! To see t h a t let V be a neighborhood of A. contained in Since U Also since B is an a - n e t w o r k Therefore A c UV1B c a. It _< of V and UNB E /1riB, follows that aanw(X) _< • X is consists let of X, t h e n A c U for some U E G.

### A calculus for branched spines of 3-manifolds by Francesco Costantino

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