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The purpose of this paper is to generalize a theorem of J. F. Adams. He showed in [A] that ifX is a subcomplex of an aspherical 2-complex and the fundamental groupG ofX has no non-trivial perfect subgroups, thenX is aspherical. We weaken the hypothesis onG to “no infinite perfect subgroups”.

Let G G be a finite group. We use the fact that each element of the Whitehead group Wh ( G ) (G) may be represented by at most a 2 × 2 2 \times 2 (nonsingular) matrix to deduce results about when simple homotopy type and homotopy type agree. As examples, we give complete descriptions of the simple homotopy types for ( Z m × Z n , 2 ) ({Z_m} \times {Z_n},2) -complexes, provided S K 1 ( Z ( Z m × Z n ) ) = 0 S{K_1}(Z({Z_m} \times {Z_n})) = 0 .

In this note we study a certain class of groups π \pi for which the homotopy classification of ( π , m ) (\pi ,m) -complexes is independent of the k k -invariant for small m m .

A 2 2 -type is a triple ( π , π 2 , k ) (\pi ,{\pi _2},k) , where π \pi is a group, π 2 {\pi _2} a π \pi -module and k ∈ H 3 ( π , π 2 ) k \in {H^3}(\pi ,{\pi _2}) . The following question is studied: When is a 2 2 -type ( π , π 2 , k ) (\pi ,{\pi _2},k) realizable by 2 2 -dimensional CW-complex X X such that the 2 2 -type ( π 1 X , π 2 X , k ( X ) ) ({\pi _1}X,{\pi _2}X,k(X)) is equivalent to ( π , π 2 , k ) (\pi ,{\pi _2},k) ? A long list of necessary conditions is given (2.2). One necessary and sufficient condition (3.1) is proved, provided π \pi has the property that stably free, finitely generated π \pi -modules are free. “Stable” 2 2 -realizability is characterized (4.1) in terms of the Wall invariant of [15]. Finally, techniques of [5] are used to extend C. T. C. Wall’s Theorem F of [15] to a space X X which is dominated by a finite CW-complex of dimension 2, provided π 1 X {\pi _1}X is finite cyclic. Under these conditions X X has the homotopy type of a finite 2 2 -complex if and only if the Wall invariant vanishes.

A connected, finite two-dimensional CW-complex with fundamental group isomorphic toG is called a [G, 2] f -complex. LetL⊲G be a normal subgroup ofG. L has weightk if and only ifk is the smallest integer such that there exists {l1,…,lk}⊆L such thatL is the normal closure inG of {l1,…,lk}. We prove that a [G, 2] f -complexX may be embedded as a subcomplex of an aspherical complexY=X∪{e 1 2 ,…,e k 2 } if and only ifG has a normal subgroupL of weightk such thatH=G/L is at most two-dimensional and defG=defH+k. Also, ifX is anon-aspherical [G, 2] f -subcomplex of an aspherical 2-complex, then there exists a non-trivial superperfect normal subgroupP such thatG/P has cohomological dimension ≤2. In this case, any torsion inG must be inP.

In (5),I considered two-term conditions in π-exact couples, of which the exact couple of Federer (7) is an example. Let M(X, Y)be the space of all maps from X to Y with the compact-open topology. Our aim in this paper is to construct a π-exact couple , where Xis a finite-dimensional (in the sense of Lebesgue) metric space and , a certain (rather large) class of spaces. Specifically, is the class of all topological spaces Xwhich possess the following property (P).(P) Let Y be a (possibly infinite) simplicial complex. There exists x0 ∈ X and y0 ∊ Y such that [X, x0]≃ [Y, y0].In § 5 it will be seen that contains all CW complexes and all metric absolute neighbourhood retracts (ANR)s.

In achieving his celebrated results on the homology groups of fibre spaces, J. P. Serre used the exact couple of a fibring defined by J. Leray. One of his main tools was the so-called two-term condition on the E2 term of this exact couple, which, if satisfied, yielded exact sequences, such as those of Gysin and Wang (see (5), Chapter IX). H. Fédérer, in (3), defined an exact couple (X, F, v) on the mapping space M(X, Y) = {ƒ:X → Y|X, Y are spaces and ƒ is continuous} with the compact-open topology, where X is a locally finite CW complex and Y is arc-connected and n-simple for all n.

This paper is concerned with the homotopy theory and simple homotopy theory of connected finite 2 dimensional CW complexes with finite cyclic fundamental group. The main theorem presents a complete classification of such complexes up to homotopy type, and this theorem has the corollary that homotopy type and simple homotopy type coincide for these complexes. This work is motivated by the general problem of describing the sets HT(n) and SHT(n) of homotopy types and simple homotopy types of n-complexes, that is, connected finite 2 dimensional CW-complexes with a given fundamental group n. A visually satisfying description of HT (n) or SHT (n) is that of either set as a graph whose edges connect the type of each n-complex X to the type of its sum X v S 2 with the 2-sphere S 2. These graphs are actually trees; they clearly contain no circuits, and they are connected because any two n-complexes have the same type once each is summed with an appropriate number of copies of the 2-sphere S 2. To re establish this latter observation ofJ. H. C. Whitehead ([20, Theorem 12]), note that each n-complex has the simple homotopy type of one modeled in an obvious fashion on some finite presentation of the fundamental group ~z (see Proposition 1). But two finite presentations of the same group n differ by a finite sequence of Tietze operations, two of which leave the simple homotopy type of the associated topological model unchanged, while two alter the simple homotopy type by an S 2 summand. Of special interest in each of these trees are the roots and the junctions. The roots are the (simple) homotopy types that do not admit a factorization involving an S 2 summand; they generate the rest of the types in the tree under the operation of forming sum with S 2. The junctions are the (simple) homotopy types that admit two or more inequivalent factorizations involving an S 2 summand; they determine the shape of the tree. Each junction is a 2-dimensional instance of non-cancellation of the 2-sphere S 2 with respect to the sum operation. When the group n is a free group F of finite rank or is the finite cyclic group Z~ of prime order q, complete descriptions of the trees HT (n) and SHT (n) can be derived from the literature, as follows. A result of C. T. C. Wall ([17, Proposition 3.3]) can be specialized to read that for a free group F of finite rank r every F-complex has the homotopy type of a sum of r

Let π \pi be Z m × Z n {Z_m}\, \times \,{Z_n} . In this paper the homotopy types of finite connected two dimensional CW-complexes with fundamental group π \pi are shown to depend only on the Euler characteristic. The basic method is to study the structure of the group Ext Z π 1 ( I π 2 , Z ) {\text {Ext}}_{Z\pi }^1(I{\pi ^2},\,Z) as a principal End( I π 2 ) {\text {End(}}I{\pi ^2}{\text {)}} -module.