An obstruction to finiteness of $CW$-complexes
An obstruction to finiteness of $CW$-complexes
复制标题
$CW$ 复合体有限性的障碍
DOI：
10.1090/s0002-9904-1964-11114-9
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发表时间：
1964
影响因子：
1.3
通讯作者：
C. Wall
中科院分区：
文献类型：
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作者：
C. Wall
单元结构是描述空间的一种方便的手段;因此，在可能的情况下，将这种结构简化为更简单的结构是很重要的。例如，一个紧致拓扑流形（或更一般的ANR）是否具有有限CW-复形的同伦类型仍然没有解决。根据Milnor [2]，这可以从猜想得出，任何被有限复形支配的CW-复形都具有有限复形的同伦类型，但我们在下面证明这是错误的。设X是连通CW-复形，具有泛覆盖X，基本群w具有（整）群环A.考虑以下条件：（i）X被有限型复形支配（即，每个维具有有限数目的单元），（ii）T和所有Hi（X）是可数的，（iii）Ar对于N<i，Hi（X）=0，并且对于所有系数丛（B）（在Steenrod意义上;如果 * = 2，则推广到非阿贝尔系数），Hi（X）=0，（B）= 0。我们的结果如下：（A）如果（i）成立，则X同伦等价于有限型复形。(B)若A是Noether模，则（i）等价于：w是n-表示的，且所有Hi（X）都是n-生成的A-模。(C)如果X被可数复形支配，它是同伦等价于1;这个条件等价于（ii）。(E)如果（iii）成立，且iV^2，则X有N维复形的同伦类型，可数如果（ii）成立。(F)X被有限复形支配当且仅当（i）和某些（iii）AT成立。如果是这种情况，且N^2，则在投射类群J中存在障碍物Q（X）？°（A），它只依赖于X的同伦类型，并且对于X有限为零。若0（X）= 0，则X具有维数为max（3，N）的有限复形的同伦型。对于iV^2，任何维数为N的有限复形K，和a ∈ j ∈（7 Ti（i ∈）），存在一个复形X，其K的（N-l）-型满足（i）和（iii）iv，且θ（X）=a。证明主要是通过归纳法;我们得到复形K和r-连通映射</>：K->X，其中K在（A）中有限，在（C）中可数。然后，我们证明了7 rr +i（0）是在（A）中（在A上）生成的，
A cell structure is a convenient means of describing a space; thus it is important to reduce such a structure to a simpler one when possible. For example, it remains unsolved whether a compact topological manifold (or more generally, ANR) has the homotopy type of a finite CW-complex. According to Milnor [2], this would follow from the conjecture that any CW-complex which is dominated by a finite complex has the homotopy type of a finite complex, but we show below that this is false. Let X be a connected CW-complex, with universal cover X, and fundamental group w with (integral) group ring A. Consider the following conditions: (i) X is dominated by a complex of finite type (i.e., one with a finite number of cells of each dimension), (ii) T and all Hi(X) are countable, (iii)Ar For N<i, Hi(X)=0 and #*(X;(B) = 0 for all coefficient bundles (B (in the sense of Steenrod ; generalised to non-abelian coefficients if * = 2). Our results are as follows: (A) If (i) holds, X is homotopy equivalent to a complex of finite type. (B) If A is noetherian, (i) is equivalent to: w is finitely presented, and all Hi(X) are finitely generated A-modules. (C) If X is dominated by a countable complex, it is homotopy equivalent to one; this condition is equivalent to (ii). (E) If (iii)isr holds, and iV^2, X has the homotopy type of an Ndimensional complex, countable if (ii) holds. (F) X is dominated by a finite complex if and only if (i) and some (iii)AT hold. When this is the case, and N^2, there is an obstruction Q(X) in the projective class group J?°(A), which depends only on the homotopy type of X, and is zero for X finite. If 0(X) = 0, X has the homotopy type of a finite complex of dimension max(3, N). For iV^2 , any finite complex K of dimension N, and a£j£(7Ti(i£)), there is a complex X, with the (N— l)-type of K, satisfying (i) and (iii)iv, and with 6(X) =a. The proofs are mostly by induction; we obtain complexes K and r-connected maps </>: K—>X, where K is finite in (A), countable in (C). We then prove that 7rr+i(0) is finitely generated (over A) in (A),