View source for Hopf invariant

An invariant of a homotopy class of mappings of topological spaces. It was first defined by H. Hopf ([[#References|[1]]], [[#References|[2]]]) for mappings of spheres <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h0480001.png" />.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h0480002.png" /> be a continuous mapping. By transition, if necessary, to a homotopic mapping, one may assume that this mapping is simplicial with respect to certain triangulations of the spheres <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h0480003.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h0480004.png" />. Then the Hopf invariant is defined as the [[Linking coefficient|linking coefficient]] of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h0480005.png" />-dimensional disjoint submanifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h0480006.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h0480007.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h0480008.png" /> for any distinct <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h0480009.png" />.

The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800010.png" /> determines an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800011.png" />, and the image of the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800012.png" /> under the homomorphism

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800013.png" /></td> </tr></table>

coincides with the Hopf invariant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800014.png" /> (here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800015.png" /> is the Hurewicz homomorphism) [[#References|[3]]].

Suppose now that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800016.png" /> is a mapping of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800017.png" /> and that a form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800018.png" /> is a generator of the integral cohomology group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800019.png" />. For such a form one may take, for example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800020.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800021.png" /> is the volume element on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800022.png" /> in some metric (for example, in the metric given by the imbedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800023.png" />), and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800024.png" /> is the volume of the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800025.png" />. Then the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800026.png" /> is closed and it is exact because the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800027.png" /> is trivial. Thus, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800028.png" /> for some form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800029.png" />. A formula for the computation of the Hopf invariant is (see [[#References|[4]]]):

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The definition of the Hopf invariant can be generalized (see [[#References|[5]]], [[#References|[6]]]) to the case of mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800031.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800032.png" />. In this case there is a decomposition

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800033.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

where

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is the homomorphism induced by the projection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800035.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800036.png" /> be the mapping given by contracting the equator of the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800037.png" /> to a point. Then the Hopf invariant is defined as the homomorphism

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800038.png" /></td> </tr></table>

under which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800039.png" /> is transformed to the projection of the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800040.png" /> onto the direct summand <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800041.png" /> in the decomposition (*). Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800042.png" />, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800043.png" /> one obtains the usual Hopf invariant. The generalized Hopf invariant is defined as the composite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800044.png" /> of the homomorphisms

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800045.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800046.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800047.png" /> is the projection of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800048.png" /> onto the direct summand <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800049.png" />, and the homomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800050.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800051.png" /> are described above. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800052.png" /> the Hopf–Whitehead invariant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800053.png" /> and the Hopf–Hilton invariant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800054.png" /> are connected by the relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800055.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800056.png" /> is the [[Suspension|suspension]] homomorphism (see [[#References|[6]]]).

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800057.png" /> be a mapping and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800058.png" /> be its cylinder (cf. [[Mapping cylinder|Mapping cylinder]]). Then the cohomology space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800059.png" /> has as homogeneous <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800060.png" />-basis a pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800061.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800062.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800063.png" />. Here the relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800064.png" /> holds (see [[#References|[7]]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800065.png" /> is odd, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800066.png" /> (because multiplication in cohomology is skew-commutative).

There is (see [[#References|[8]]]) a generalization of the Hopf–Steenrod invariant in terms of a generalized cohomology theory (cf. [[Generalized cohomology theories|Generalized cohomology theories]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800067.png" /> be the semi-exact homotopy functor in the sense of Dold (see [[#References|[9]]]), given on the category of finite CW-complexes and taking values in a certain Abelian category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800068.png" />. Then the mapping of complexes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800069.png" /> determines an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800070.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800071.png" /> is the set of morphisms in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800072.png" />. The Hopf–Adams invariant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800073.png" /> is defined when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800074.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800075.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800076.png" /> is the corresponding suspension mapping. In this case the sequence of cofibrations

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800077.png" /></td> </tr></table>

corresponds to an exact sequence in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800078.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800079.png" /></td> </tr></table>

which determines the Hopf–Adams–Steenrod invariant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800080.png" />.

In the case of the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800081.png" /> taking values in the category of modules over the [[Steenrod algebra|Steenrod algebra]] modulo 2, one obtains the Hopf–Steenrod invariant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800082.png" /> of a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800083.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800084.png" /> (see [[#References|[7]]]). The cohomology space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800085.png" /> has as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800086.png" />-basis a pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800087.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800088.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800089.png" />, and then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800090.png" /></td> </tr></table>

The Hopf invariant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800091.png" /> modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800092.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800093.png" /> is a prime number) is defined as the composite of the mappings

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800094.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800095.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800096.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800097.png" /> is the localization by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800098.png" /> of the pair of spaces (see [[#References|[10]]]). Let

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800099.png" /></td> </tr></table>

be the suspension homomorphism. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h048000100.png" /> (see [[#References|[10]]]). The Hopf invariant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h048000101.png" /> can also be defined in terms of the Stiefel numbers (cf. [[Stiefel number|Stiefel number]]) (see [[#References|[11]]]): If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h048000102.png" /> is a closed equipped manifold and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h048000103.png" />, then the characteristic Stiefel–Whitney number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h048000104.png" /> of the normal bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h048000105.png" /> is the same as the Hopf invariant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h048000106.png" /> of the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h048000107.png" /> that is a representative of the class of equipped cobordisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h048000108.png" />.

The Adams–Novikov spectral sequence makes it possible to construct higher Hopf invariants. Namely, one defines inductively the invariants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h048000109.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h048000110.png" /> (see [[#References|[12]]]). From the form of the differentials of this spectral sequence it follows that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h048000111.png" /></td> </tr></table>

(where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h048000112.png" /> is the ring of complex point cobordisms); therefore, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h048000113.png" />, the invariants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h048000114.png" /> lie in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h048000115.png" /> and are called the Hopf–Novikov invariants. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h048000116.png" /> one obtains the Adams invariant.

The values that a Hopf invariant can take are not arbitrary. For example, for a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h048000117.png" /> the Hopf invariant is always 0. The Hopf invariant modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h048000118.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h048000119.png" />, is trivial, except when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h048000120.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h048000121.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h048000122.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h048000123.png" />. On the other hand, for any even number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h048000124.png" /> there exists a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h048000125.png" /> with Hopf invariant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h048000126.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h048000127.png" /> is arbitrary). For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h048000128.png" /> there exists mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h048000129.png" /> with Hopf invariant 1.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Hopf,   "Ueber die Abbildungen der dreidimensionalen Sphäre auf die Kügelfläche"  ''Math. Ann.'' , '''104'''  (1931)  pp. 639–665</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Hopf,   "Ueber die Abbildungen von Sphären niedriger Dimension"  ''Fund. Math.'' , '''25'''  (1935)  pp. 427–440</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J.-.P. Serre,   "Groupes d'homotopie et classes de groupes abéliens"  ''Ann. of Math.'' , '''58''' :  2  (1953)  pp. 258–294</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  J.H.C. Whitehead,   "An expression of the Hopf invariant as an integral"  ''Proc. Nat. Acad. Sci. USA'' , '''33'''  (1937)  pp. 117–123</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  J.H.C. Whitehead,   "A generation of the Hopf invariant"  ''Ann. of Math. (2)'' , '''51'''  (1950)  pp. 192–237</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  P. Hilton,   "Suspension theorem and generalized Hopf invariant"  ''Proc. London. Math. Soc. (3)'' , '''1''' :  3  (1951)  pp. 462–493</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  N. Steenrod,   "Cohomologies invariants of mappings"  ''Ann. of Math. (2)'' , '''50'''  (1949)  pp. 954–988</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  J. Adams,   "On the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h048000130.png" />"  ''Topology'' , '''5'''  (1966)  pp. 21–71</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  A. Dold,   "Halbexakte Homotopiefunktoren" , Springer  (1966)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  D. Husemoller,   "Fibre bundles" , McGraw-Hill  (1966)</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  R.E. Stong,   "Notes on cobordism theory" , Princeton Univ. Press  (1968)</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top">  S.P. Novikov,   "The methods of algebraic topology from the view point of cobordism theories"  ''Math. USSR-Izv.'' , '''4''' :  1  (1967)  pp. 827–913  ''Izv. AKad. Nauk SSSR Ser. Mat.'' , '''31''' :  4  (1967)  pp. 855–951</TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top">  J.F. Adams,   "On the non-existence of elements of Hopf invariant one"  ''Ann. of Math.'' , '''72'''  (1960)  pp. 20–104</TD></TR></table>