## Doctoral Theses

### The Fixed Point Index as a Local Lefschetz Number.

2-28-1996

2-28-1997

#### Institute Name (Publisher)

Indian Statistical Institute

Doctoral Thesis

#### Degree Name

Doctor of Philosophy

Mathematics

#### Department

Theoretical Statistics and Mathematics Unit (TSMU-Kolkata)

Mukherjee, K. K.

#### Abstract (Summary of the Work)

In this thesis we defitne a class of self maps of connected compact polyhodza - those which prmserve erpanding directions - and define the fixed point indices of such maps at an isolated set of fixed points of the map as a local Lefschets rumber. Our definition uses simplicial approximations of the given map in the spirit of O Nell (I19| and Fournier (71) and is intrinsic so that it is computable.Let X be a connected compact polyhedron and f:Xâ†’ X be a map an X. The Lefscheta number L() of / is then defined to be ([13]),L) -E(-1)jTrace {, : H2(X,Q) - H2(x,Q)}By a Lefschota number we will always mean the alternating sam of the traces of a chain (cocbain) map on a chain (cochnin) complex and not Just the corresponding uumber of a graded endomorphism. The fixod point set of / is,Fix f (a â‚¬ X: f(a) - )Theorem (The Lefschetz Fixed Point Theorem, (14), (16]) If L(S) +0 the fired point aet of ang map homotopie to f ia nonempty. Let U be an open subeet of X such that an Fix f = 0, where 8U is the frontier of U in x. Then,Theorem It is possible to canonically associate un integer i(.U) swith U called the fired point indez of f at U, uhich can be cheracter- ieed by a sct of simple artoma, (see Brown (4), or Dold (3). (6).Definition A set of fixed points C c Fix f is called an iaolated net of fixed points if C is compact and open in Fix f.Let C be an isolated set of fixed points of f and W be any open neigh- bourhood of C in |K] such that, Wn Fix / = Wn Fix f = C. Then the fixed point inder i(f.C), of f at C is defined to be, ilf, C) = i(f, W) (aee Jiang [12). This definition of (,C) ls independant of the cholce of the open neighbourhood of C.Since Fix f is a compact subset of X, it is clear that any collection of distinct isolated sets of fixed points of f is finite.Theorem Let C1...Ck, be a collecfion of isolated sets of fired points of f such that Firf -EG, Then, un - L4.C). (*) If one can express i(f, C) explicitiy in terms of local data conceraing f in a neighbourhood of C, then (+) is called a Lefschetz Fized Point Formula or LFPF for short. Of particular interest is the case when the map f has finitely many isolated fixed points, say (P): then i(f, ()) is thought of as the algebraic multiplicity of the fixed point p.For a smooth map :Xâ†’X which is transverse to the diagonal, the index at a trasversal fixed point is the sign of det(1- f.(Pk)), (see |1]) where f.(p) : Xpâ†’ Xp is the derivative of f at the fixed point p and 1 is the identity map. (This is the simplest aud oldest example of a LFPF). This farmala can be casily reorganised so as to exhibit the index ilf.p) as a local Lefschets Number.

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