In 113] I indicated a proof of the following theorem: THEOREM A. Let w be a finite group of order n. Let d be the greatest common divisor of n and p(n), p being Euler's p-function. Suppose 7r has periodic cohomology of period q. Then there exists a finite simplicial complex X of dimension dq -1 which has the homotopy type of a (dq -1)sphere and on which w acts freely' and simplicially. I do not know whether it is always possible to replace dq 1 by q 1 in this Theorem. This is, of course, the best possible value [3, Ch. XVI, ? 9]. This replacement is possible in special cases. In the appendix, I will consider the case wr 33, the symmetric group on three letters and will show that S3 can act freely and simplicially on a finite 3-dimensional homotopy 3-sphere X. The period of 33 is 4 so this is a best possible result. Note that a result of Milnor [71 shows that for some groups r, X cannot be a manifold. In particular, Milnor's result shows that 33 cannot act freely on any closed manifold with the mod 2 homology of a sphere. Consequently, the space X which I will construct in the appendix cannot be a manifold and so cannot be homeomorphic to S3. I will give here a complete proof of Theorem A as well as the proof of a "mod C" generalization. Before stating this generalization, it is necessary to give some definitions. Let 7C be any finite group. Let 9? be any set of primes. Then there is a 9f-period (possibly infinite) associated with the group r [3, Ch. XII, Ex. 11]. This is just the ordinary period if f? is the set of all primes. For any set of primes ?1?, let C?I2 be the Serre class [10] of finite abelian groups having no p-torsion for any prime p e 9?.

Periodic Resolutions for Finite Groups

Recent developments in differential and PL-topology have succeeded in reducing a large number of problems (classification and embedding, for example) to problems in homotopy theory. The classical methods of homotopy theory are available for these problems, but are often not strong enough to give the results needed. In this paper we attempt to develop a branch of homotopy theory applicable to the classification problem for compact manifolds. A Poincare complex is (approximately) a finite cw-complex which satisfies the Poincare duality theorem. A precise definition is given in ? 1, together with a discussion of chain complexes. In Chapter 2, we give a cutting and gluing theorem, define connected sum, and give a theorem on product decompositions. Chapter 3 is devoted to an account of the tangential properties first introduced by M. Spivak (Princeton thesis, 1964). We then start our classification theorems; in Chapter 4, for dimensions up to 3, where the dominant invariant is the fundamental group; and in Chapter 5, for dimension 4, where we obtain a classification theorem when the fundamental group has prime order. It is complicated to use, but allows us to construct two interesting examples. In the second part of this paper, we intend to classify highly connected Poincare complexes; to show how to perform surgery, and give some applications; by constructing handle decompositions and computing some cobordism groups. This paper was originally planned when the only known fact about topological manifolds (of dimension >3) was that they were Poincare complexes. Novikov's proof [301 of topological invariance of rational Pontrjagin classes and subsequent work in the same direction has changed this, but we can still easily summarize the basis of the relation of Poincare complexes to smooth and PL-manifolds. The first point of difference lies in the structure group of the normal bundle; G, 0, or PL in the three cases. If this group be appropriately reduced (from G to 0 or PL), surgery can be performed as in [13] to try to construct a manifold; certain algebraic obstructions arise in the middle dimension (see [13] for the simply-connected and [26] for the general case). These algebraic structures provide the second point of difference; one in which we are particularly interested. Chapter 5 was originally written for the purpose of constructing examples to illustrate these (5.4.1 and 5.4.2). A fuller discussion of these points is planned to appear as a sequel to [26].

Poincare Complexes: I

For dimensions greater than five the main theorem gives necessary and sufficient conditions that a smooth open manifold W be the interior of a smooth compact manifold with boundary. The basic necessary condition is that each end ! of W be tame. Tameness consists of two parts (a) and (b): (a) The system of fundamental groups of connected open neighborhoods of ! is stable. This means that (with any base points and connecting paths) there exists a cofinal sequence G1 G2 f1 !! . . . f2 !! so that isomorphisms are induced Image(f1) Image(f2) ∼= !! . . . ∼= !! . (b) There exist arbitrarily small open neighborhoods of ! that are dominated each by a finite complex. Tameness for ! clearly depends only on the topology of W . It is shown that if W is connected and of dimension ! 5, its ends are all tame if and only if W ×S is the interior of a smooth compact manifold. However examples of smooth open manifolds W are constructed in each dimension ! 5 so that W itself is not the interior of a smooth compact manifold although W ×S is. When (a) holds for ! , the projective class group K̃0(π1(!)) = lim ←− j Gj is well defined up to canonical isomorphism. When ! is tame an invariant σ(!) ∈ K̃0(π1!) is defined using the smoothness structure as well as the topology of W . It is closely related to Wall’s obstruction to finiteness for CW complexes (Annals of Math. 81 (1965) pp. 56-69). Copyright declaration is printed here 2 Laurence C. Siebenmann Main Theorem. A smooth open manifold W , n > 5, is the interior of a smooth compact manifold if and only if W has finitely many connected components, and each end ! of W is tame with invariant σ(!) = 0. (This generalizes a theorem of Browder, Levine, and Livesay, A.M.S. Notices 12, Jan. 1965, 619-205). For the study of σ(!), a sum theorem and a product theorem are established for C.T.C. Wall’s related obstruction. Analysis of the different ways to fit a boundary onto W shows that there exist smooth contractible open subsets W of R , n odd, n > 5, and diffeomorphisms of W onto itself that are smoothly pseudo-isotopic but not smoothly isotopic. The main theorem can be relativized. A useful consequence is Proposition. Suppose W is a smooth open manifold of dimension > 5 and N is a smoothly and properly imbedded submanifold of codimension k %= 2. Suppose that W and N separately admit completions. If k = 1 suppose N is 1-connected at each end. Then there exists a compact manifold pair (W, N) such that W = IntW , N = IntN . If W is a smooth open manifold homeomorphic to M × (0, 1) where M is a closed connected topological (n − 1)-manifold, then W has two ends !− and !+ , both tame. With π1(!−) and π1(!+) identified with π1(W ) there is a duality σ(!+) = (−1)σ(!−) where the bar denotes a certain involution of the projective class group K̃0(π1W ) analogous to one defined by J.W. Milnor for Whitehead groups. Here are two corollaries. If M is a stably smoothable closed topological manifold, the obstruction σ(M) to M having the homotopy type of a finite complex has the symmetry σ(M) = (−1)σ(M). If ! is a tame end of an open topological manifold W and !1 , !2 are the corresponding smooth ends for two smoothings of W , then the difference σ(!1)−σ(!2) = σ0 satisfies σ0 = (−1)σ0 . Warning: In case every compact topological manifold has the homotopy type of a finite complex all three duality statements above are 0 = 0. It is widely believed that all the handlebody techniques used in this thesis have counterparts for piecewise-linear manifolds. Granting this, all the above results can be restated for piecewise-linear manifolds with one slight exception. For the proposition on pairs (W, N) one must insist that N be locally unknotted in W in case it has codimension one.

The obstruction to finding a boundary for an open manifold : of dimension greater than five

To complete the classification of self-maps of BG (Theorem 4.2), it remains to prove Theorem 4.1, i.e., to show that the higher limits of certain functors n 1H 2: f p(G) Z(p-Wall vanish and that H0(Mp(G); f i) has the expected value. In this section, these limits H*(Mp(G); Hi) will be calculated in the "easy" cases, when G is a simply connected classical group (i.e., one of the groups SU(n), Sp(n), or Spin(n)), or when i = 2. The computation of H*(Mp(G); Hi) for the simply connected exceptional Lie groups will be put off until Section 6. We begin with some general properties of higher limits. Let 6 be any small discrete category. It is convenient, when dealing with higher limits of functors on -, to let 4-moldenote the category of contravariant functors from v to Ad. Then for any F e W-muok, we can identify

https://www.jstor.org/stable/pdfplus/2946589.pdf

Homotopy classification of self-maps of BG via G-actions

Introduction 1. Extensions and cross sections 1.1. Extensions 1.2. Cross sections 2. Nakayama's duality 2.1. Preliminaries 2.2. Nakayama's duality 2.3. Some sequences 3. The numbers T(T), h(T) and i(T) 3.1. Functorial properties 3.2. The number r(T) 3.3. The number h(T) 3.4. The number i(T) 4. The numbers T(E), h(E) and i(E) 4.1. A function f (E) 4.2. The number T(E) 4.3. Proof of (Al), (A2) 4.4. Relation among r(E), h(E) and i(E) 4.5. Cyclic splitting field 5. Main theorem 6. Examples 6.1. The sequence (N) 6.2. Construction of T with T(T) E Z 6.3. An example of semi-simple group G with z(G) E Z References

https://www.jstor.org/stable/pdfplus/1970502.pdf

On the tamagawa number of algebraic tori

Modules Over Finite Groups

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Modules Over Finite Groups

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Periodic Resolutions for Finite Groups

Poincare Complexes: I

The obstruction to finding a boundary for an open manifold : of dimension greater than five

Homotopy classification of self-maps of BG via G-actions

On the tamagawa number of algebraic tori

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