Involutions on Manifolds by Dr. Santiago López de Medrano (auth.)

By Dr. Santiago López de Medrano (auth.)

This ebook comprises the result of paintings performed in the course of the years 1967-1970 on fixed-point-free involutions on manifolds, and is an enlarged model of the author's doctoral dissertation [54J written below the course of Professor William Browder. the topic of fixed-paint-free involutions, as a part of the topic of staff activities on manifolds, has been a major resource of difficulties, examples and ideas in topology for the final 4 many years, and gets renewed consciousness each time a new technical improvement indicates new questions and strategies ([62, eight, 24, 63J). right here we ponder generally these homes of fixed-point-free involutions that may be top studied utilizing the innovations of surgical procedure on manifolds. This method of the topic used to be initiated by way of Browder and Livesay. specific realization is given the following to involutions of homotopy spheres, yet even for this actual case, a extra basic idea is particularly priceless. vital similar subject matters that we don't contact listed below are these of involutions with mounted issues, and the connection among fixed-point-free involutions and loose Sl-actions. For those subject matters, the reader is observed [23J, and to [33J, [61J, [82J, respectively. the 2 major difficulties we assault are these of type of involutions, and the lifestyles and specialty of invariant submanifolds with sure houses. As should be obvious, those difficulties are heavily similar. If (T, l'n) is a fixed-point-free involution of a homotopy sphere l'n, the quotient l'n/Tis known as a homotopy projective space.

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Two homotopy triangulations of M" are equivalent if, and only if, they have the same normal invariant and the same BrowderLivesay invariant. Proof. (/1) and q(/O)=a{fl). From the first assumption it follows that there is a normal cobordism F: W ~ M between fo and fl. 2). If n = 4k + 1 the same result holds, since the only obstruction is an Arf invariant, and we can change it by adding a Kervaire manifold to Walong the boundary. If n=4k+3, compose F withfo- 1 to obtain a normal cobordismfo- 1 F between 1: Mo~Mo andfo- 1A.

And f1 can be defined as follows: an element x determines a prefered class of immersions x: Sk--+N and a lifting x: Sk--+N2k. (x, y)=x· y+(x· T*y) t, f1(x) = <{J(x) + tf/(x) t. 3, except that we do not assume that the representative sphere i~ embedded (this is possible if, and only if, <{J(x)=O), but the definition makes sense for an immersed sphere. 34 III Relations with Non-simply-connected Surgery Obstructions Surgery can be performed if, and only if, there is a A-basis el' ... (/;) = 0 and the surgery obstruction can be identified with the algebraic obstruction to finding such a basis, which is given by index and Arf invariants.

Let M n be a manifold and NqcM n a submanifold with normal bundle (. Let fo: Mo-'> M be a normal map, and suppose fo is t-regular at N, No =fo-l(N) and go= fiNo: No-'>N the restricted normal map. Let G: Vq+l-'>N be a normal cobordism between go and gl: N1 -'> N. Then there is a normal cobordism F: wn+1-,> M between fo and 11: M 1 -'> M such that F is t-regular at N, F- 1(N)= V and FIV=G. Proof. By assumption we have bundle maps bo : VMo -'> ~ covering fo and B: vv-+(EB~IN covering G such that BlvNo coincides with the restriction of bo , vMolNoEBg~ (-+ {IN EB ( (see III.

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