Induced fibrations and cofibrations
Induced fibrations and cofibrations
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诱导纤维化和共纤维化
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发表时间：
1967
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通讯作者：
T. Ganea
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作者：
T. Ganea
导论.众所周知，任何映射都同伦等价于纤维映射，即，在纤维化中基底上的总空间的投影。然而，简单的例子表明，有一些映射不能同伦等价于全空间中纤维的任何包含，而刻画等价于这种包含的映射的问题首先在[14]中提出。通过第一点说明，可以从一开始就假设所考虑的映射是纤维映射p：E -B，因此问题被转换成表征等效于诱导纤维化的纤维化p的问题。有两个直接必要条件：p的纤维F必须具有某个环空间f1 Y的同伦型，对任意空间X，包含i：F -* E必须将广义同伦群7 r（X，fQF）映射到n（X，QE）的中心;后者是对纤维化E -* B -* Y的同伦序列中的边界算子映射XT ~ 2（Y）这一著名事实的温和推广在7 r1（E）的中心。在这个方向上的第一个结果是由于塞尔和断言，p是一个诱导纤维化，如果B是1-连通和F有一个单一的非零（阿贝尔）同伦群。这个结果被推广到[6]，[12]，[17]，[15]，使得F在连续维上至多有m-1个非零同伦群，只要它具有环空间的同伦型，并且B是（m -）-连通的.在本文的前两节中，我们给出了在满足一定条件的同伦等价下允许F在连续维数上有2 m-1个非零同伦群的结果，条件是B是（m1）-连通的，并且F具有环空间的同伦型，其中条件涉及Q B在F上的“运算”.当F在连续维数上有m个非零同伦群时，用某个Whitehead积的零化来表示诱导p的充分条件，这个结果回答了[15]中提出的一个问题。对偶地，任何映射都同伦等价于具有同伦扩张性质的包含A -> X，但很少有映射等价于通过收缩到这样一个点X的子集A而得到的识别映射。这就引出了刻画由空间Y到A的映射所诱导的上纤维化A -> X的问题。我们的结果扩展了一些以前的结果[8]，[12]的适用范围，通过施加一个额外的条件，该条件涉及悬浮EA对从X通过收缩A到一个点而获得的上纤维C的“合作”;该条件方便地用上纤维化的Hopf不变量表示
Introduction. It is well known that any map is homotopically equivalent to a fiber map, i.e., to the projection of the total space on the base in a fibration. Simple examples, however, reveal that there are maps which fail to be homotopically equivalent to any inclusion of a fiber in the total space, and the problem of characterizing the maps which are equivalent to such inclusions was first raised in [14]. By the first remark, the map under consideration may be assumed from the beginning to be a fiber map p: E -B, and the problem is thus converted into that of characterizing the fibrations p which are equivalent to induced fibrations. There are two immediate necessary conditions: the fiber F of p must have the homotopy type of some loop space fl Y, and the inclusion i: F -* E must map the generalized homotopy group 7r(X, fQF) into the center of n(X, QE) for any space X; the latter is a mild generalization of the well known fact that the boundary operator in the homotopy sequence of a fibration E -* B -* Y maps XT2( Y) into the center of 7r1(E). The first result in this direction is due to Serre and asserts that p is an induced fibration if B is 1-connected and F has a single nonvanishing (Abelian) homotopy group. This result was generalized [6], [12], [17], [15] to allow F to have at most m -1 nonvanishing homotopy groups in consecutive dimensions provided it has the homotopy type of a loop space and B is (m -)-connected. In the first two sections of this paper we give results which allow F to have 2m -1 nonvanishing homotopy groups in consecutive dimensions provided B is (m 1)-connected and F has the homotopy type of a loop space under a homotopy equivalence fulfilling a certain condition which involves the "operation" of QB on F. In case F has m nonvanishing homotopy groups in consecutive dimensions, the sufficient condition in order that p be induced is expressed by means of the vanishing of a certain Whitehead product; this result answers a question raised in [15]. Dually, any map is homotopically equivalent to an inclusion A -> X having the homotopy extension property, but few maps are equivalent to identification maps resulting by shrinking to a point such a subset A of X. This leads to the problem of characterizing the cofibrations A -> X which are induced by maps of some space Y into A. Our results extend the range of applicability of some previous results [8], [12] by imposing an additional condition which involves the "cooperation" of the suspension EA on the cofiber C obtained from X by shrinking A to a point; the condition is conveniently expressed in terms of the Hopf invariant of a cofibration