Difference between revisions of "Hopf invariant"
(Importing text file) |
Ulf Rehmann (talk | contribs) m (tex encoded by computer) |
||
Line 1: | Line 1: | ||
− | + | <!-- | |
+ | h0480001.png | ||
+ | $#A+1 = 129 n = 1 | ||
+ | $#C+1 = 129 : ~/encyclopedia/old_files/data/H048/H.0408000 Hopf invariant | ||
+ | Automatically converted into TeX, above some diagnostics. | ||
+ | Please remove this comment and the {{TEX|auto}} line below, | ||
+ | if TeX found to be correct. | ||
+ | --> | ||
− | + | {{TEX|auto}} | |
+ | {{TEX|done}} | ||
− | + | 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 $ f: S ^ {2n - 1 } \rightarrow S ^ {n} $. | |
− | + | Let $ f: S ^ {2n - 1 } \rightarrow S ^ {n} $ | |
+ | 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 $ S ^ {n} $ | ||
+ | and $ S ^ {2n - 1 } $. | ||
+ | Then the Hopf invariant is defined as the [[Linking coefficient|linking coefficient]] of the $ ( n - 1) $- | ||
+ | dimensional disjoint submanifolds $ f ^ { * } ( a) $ | ||
+ | and $ f ^ { * } ( b) $ | ||
+ | in $ S ^ {2n - 1 } $ | ||
+ | for any distinct $ a, b \in S ^ {n} $. | ||
− | + | The mapping $ f: S ^ {2n - 1 } \rightarrow S ^ {n} $ | |
+ | determines an element $ [ f] \in \pi _ {2n - 1 } ( S ^ {n} ) $, | ||
+ | and the image of the element $ [ f] $ | ||
+ | under the homomorphism | ||
− | + | $$ | |
+ | \pi _ {2n - 1 } ( S ^ {n} ) = \ | ||
+ | \pi _ {2n - 2 } ( \Omega S ^ {n} ) \rightarrow ^ { h } \ | ||
+ | H _ {2n - 2 } ( \Omega S ^ {n} ) = \mathbf Z | ||
+ | $$ | ||
− | + | coincides with the Hopf invariant $ H ( f ) $( | |
+ | here $ h $ | ||
+ | is the Hurewicz homomorphism) [[#References|[3]]]. | ||
− | + | Suppose now that $ f: S ^ {2n - 1 } \rightarrow S ^ {n} $ | |
+ | is a mapping of class $ C ^ {2} $ | ||
+ | and that a form $ \Omega \in \Lambda ^ {n} S ^ {n} $ | ||
+ | is a generator of the integral cohomology group $ H ^ {n} ( S ^ {n} , \mathbf Z ) $. | ||
+ | For such a form one may take, for example, $ \Omega = ( dV)/( \mathop{\rm vol} S ^ {n} ) $, | ||
+ | where $ dV $ | ||
+ | is the volume element on $ S ^ {n} $ | ||
+ | in some metric (for example, in the metric given by the imbedding $ S ^ {n} \subset \mathbf R ^ {n + 1 } $), | ||
+ | and $ \mathop{\rm vol} S ^ {n} $ | ||
+ | is the volume of the sphere $ S ^ {n} $. | ||
+ | Then the form $ f ^ { * } ( \Omega ) \in \Lambda ^ {n} S ^ {2n - 1 } $ | ||
+ | is closed and it is exact because the group $ H ^ {n} ( S ^ {2n - 1 } , \mathbf Z ) $ | ||
+ | is trivial. Thus, $ f ^ { * } ( \Omega ) = d \theta $ | ||
+ | for some form $ \theta \in \Lambda ^ {n - 1 } S ^ {2n - 1 } $. | ||
+ | A formula for the computation of the Hopf invariant is (see [[#References|[4]]]): | ||
− | + | $$ | |
+ | H ( f ) = \ | ||
+ | \int\limits _ {S ^ {2n - 1 } } | ||
+ | \theta \wedge d \theta . | ||
+ | $$ | ||
+ | |||
+ | The definition of the Hopf invariant can be generalized (see [[#References|[5]]], [[#References|[6]]]) to the case of mappings $ f: S ^ {m} \rightarrow S ^ {n} $ | ||
+ | for $ m \leq 4n - 4 $. | ||
+ | In this case there is a decomposition | ||
+ | |||
+ | $$ \tag{* } | ||
+ | \pi _ {m} ( S ^ {n} \lor S ^ {n} ) = \ | ||
+ | \pi _ {m} ( S ^ {n} ) \oplus | ||
+ | \pi _ {m} ( S ^ {n} ) \oplus | ||
+ | \pi _ {m} ( S ^ {2n - 1 } ) \oplus | ||
+ | \mathop{\rm ker} k _ {*} , | ||
+ | $$ | ||
where | where | ||
− | + | $$ | |
+ | k _ {*} : \pi _ {m + 1 } | ||
+ | ( S ^ {n} \times S ^ {n} ,\ | ||
+ | S ^ {n} \lor S ^ {n} ) \rightarrow \ | ||
+ | \pi _ {m + 1 } ( S ^ {2n} ) | ||
+ | $$ | ||
− | is the homomorphism induced by the projection | + | is the homomorphism induced by the projection $ k: ( S ^ {n} \times S ^ {n} , S ^ {n} \lor S ^ {n} ) \rightarrow ( S ^ {n} , \mathop{\rm pt} ) $. |
+ | Let $ g: S ^ {n} \rightarrow S ^ {n} \lor S ^ {n} $ | ||
+ | be the mapping given by contracting the equator of the sphere $ S ^ {n} $ | ||
+ | to a point. Then the Hopf invariant is defined as the homomorphism | ||
− | + | $$ | |
+ | H: \pi _ {m} ( S ^ {n} ) \rightarrow \pi _ {m} ( S ^ {2n - 1 } ) | ||
+ | $$ | ||
− | under which | + | under which $ [ f] \in \pi _ {m} ( S ^ {n} ) $ |
+ | is transformed to the projection of the element $ [ g \circ f] \in \pi _ {m} ( S ^ {n} \lor S ^ {n} ) $ | ||
+ | onto the direct summand $ \pi _ {m} ( S ^ {2n - 1 } ) $ | ||
+ | in the decomposition (*). Since $ \pi _ {2n - 1 } ( S ^ {2n - 1 } ) = \mathbf Z $, | ||
+ | for $ m = 2n - 1 $ | ||
+ | one obtains the usual Hopf invariant. The generalized Hopf invariant is defined as the composite $ H ^ {*} $ | ||
+ | of the homomorphisms | ||
− | + | $$ | |
+ | \pi _ {m} ( S ^ {n} ) \rightarrow ^ { {g _ *} } \ | ||
+ | \pi _ {m} ( S ^ {n} \lor S ^ {n} ) \rightarrow ^ { p } \ | ||
+ | $$ | ||
− | + | $$ | |
+ | \rightarrow ^ { p } \pi _ { m + 1 } ( S ^ {n} \times S ^ {n} | ||
+ | , S ^ {n} \lor S ^ {n} ) \rightarrow ^ { {k _ * } } \pi _ {m + 1 } ( S ^ {2n} ), | ||
+ | $$ | ||
− | where | + | where $ p $ |
+ | is the projection of the group $ \pi _ {m} ( S ^ {n} \lor S ^ {n} ) $ | ||
+ | onto the direct summand $ \pi _ {m + 1 } ( S ^ {n} \times S ^ {n} , S ^ {n} \lor S ^ {n} ) $, | ||
+ | and the homomorphisms $ g _ {*} $ | ||
+ | and $ k _ {*} $ | ||
+ | are described above. For $ m \leq 4n - 4 $ | ||
+ | the Hopf–Whitehead invariant $ H $ | ||
+ | and the Hopf–Hilton invariant $ H ^ {*} $ | ||
+ | are connected by the relation $ H ^ {*} = S \circ H $, | ||
+ | where $ S: \pi _ {m} ( S ^ {2n - 1 } ) \rightarrow \pi _ {m + 1 } ( S ^ {2n} ) $ | ||
+ | is the [[Suspension|suspension]] homomorphism (see [[#References|[6]]]). | ||
− | Let | + | Let $ f: S ^ {2n - 1 } \rightarrow S ^ {n} $ |
+ | be a mapping and let $ C _ {f} $ | ||
+ | be its cylinder (cf. [[Mapping cylinder|Mapping cylinder]]). Then the cohomology space $ H ^ {*} ( C _ {f} , S ^ {2n - 1 } ) $ | ||
+ | has as homogeneous $ \mathbf Z $- | ||
+ | basis a pair $ \{ a, b \} $ | ||
+ | with $ \mathop{\rm dim} a = n $ | ||
+ | and $ \mathop{\rm dim} b = 2n $. | ||
+ | Here the relation $ a ^ {2} = H ( f ) b $ | ||
+ | holds (see [[#References|[7]]]). If $ n $ | ||
+ | is odd, then $ H ( f ) = 0 $( | ||
+ | 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 | + | 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 $ k $ |
+ | 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 $ A $. | ||
+ | Then the mapping of complexes $ f: X \rightarrow Y $ | ||
+ | determines an element $ f ^ { * } = d ( f ) \in \mathop{\rm Hom} ( k ( Y), k ( X)) $, | ||
+ | where $ \mathop{\rm Hom} $ | ||
+ | is the set of morphisms in $ A $. | ||
+ | The Hopf–Adams invariant $ e ( f ) $ | ||
+ | is defined when $ f ^ { * } = 0 $ | ||
+ | and $ d ( Sf ) = 0 $, | ||
+ | where $ Sf: SX \rightarrow SY $ | ||
+ | is the corresponding suspension mapping. In this case the sequence of cofibrations | ||
− | + | $$ | |
+ | X \rightarrow ^ { f } \ | ||
+ | Y \rightarrow ^ { f } \ | ||
+ | Y \cup _ {f} CX \rightarrow ^ { j } \ | ||
+ | SX \mathop \rightarrow \limits ^ {-} Sf \ | ||
+ | SY | ||
+ | $$ | ||
− | corresponds to an exact sequence in | + | corresponds to an exact sequence in $ A $: |
− | + | $$ | |
+ | 0 \leftarrow k ( X) \leftarrow ^ { {i _ *} } \ | ||
+ | k ( Y \cup _ {f} CX) \leftarrow ^ { {j _ *} } \ | ||
+ | k ( SX) \leftarrow 0, | ||
+ | $$ | ||
− | which determines the Hopf–Adams–Steenrod invariant | + | which determines the Hopf–Adams–Steenrod invariant $ e ( f ) = \mathop{\rm Ext} ^ {1} ( k ( Y), k ( X)) $. |
− | In the case of the functor | + | In the case of the functor $ k = H ^ {*} ( - ; \mathbf Z _ {2} ) $ |
+ | taking values in the category of modules over the [[Steenrod algebra|Steenrod algebra]] modulo 2, one obtains the Hopf–Steenrod invariant $ H _ {2} ( f ) \in \mathbf Z $ | ||
+ | of a mapping $ f: S ^ {m} \rightarrow S ^ {n} $ | ||
+ | for $ m > n $( | ||
+ | see [[#References|[7]]]). The cohomology space $ H ^ {*} ( C _ {f} , S ^ {m} ; \mathbf Z _ {2} ) $ | ||
+ | has as $ \mathbf Z _ {2} $- | ||
+ | basis a pair $ \{ a, b \} $ | ||
+ | with $ \mathop{\rm dim} a = n $ | ||
+ | and $ \mathop{\rm dim} b = m + 1 $, | ||
+ | and then | ||
− | + | $$ | |
+ | Sq ^ {m - n + 1 } a = \ | ||
+ | H _ {2} ( f ) b. | ||
+ | $$ | ||
− | The Hopf invariant | + | The Hopf invariant $ H _ {p} $ |
+ | modulo $ p $( | ||
+ | where $ p $ | ||
+ | is a prime number) is defined as the composite of the mappings | ||
− | + | $$ | |
+ | \pi _ {2pn} ( S ^ {2n + 1 } ) _ {(} p) \mathop \rightarrow \limits ^ \approx \ | ||
+ | \pi _ {2pn - 2 } ( \Omega ^ {2} S ^ {2n + 1 } ) _ {(} p) \rightarrow | ||
+ | $$ | ||
− | + | $$ | |
+ | \rightarrow \ | ||
+ | \pi _ {2pn - 2 } ( \Omega ^ {2} S ^ {2n + 1 } , S ^ {2n - 1 } ) _ {(} p) \rightarrow | ||
+ | $$ | ||
− | + | $$ | |
+ | \rightarrow \ | ||
+ | H _ {2pn - 2 } ( \Omega ^ {2} S ^ {2n + | ||
+ | 1 } , S ^ {2n - 1 } ) _ {(} p) = \mathbf Z /p, | ||
+ | $$ | ||
− | where | + | where $ ( X, Y) _ {p} $ |
+ | is the localization by $ p $ | ||
+ | of the pair of spaces (see [[#References|[10]]]). Let | ||
− | + | $$ | |
+ | S: \pi _ {4n - 1 } ( S ^ {2n} ) \rightarrow \ | ||
+ | \pi _ {4n} ( S ^ {2n + 1 } ) | ||
+ | $$ | ||
− | be the suspension homomorphism. Then | + | be the suspension homomorphism. Then $ H _ {2} ( Sf ) = H _ {2} ( f ) $( |
+ | see [[#References|[10]]]). The Hopf invariant $ H ( f ) $ | ||
+ | can also be defined in terms of the Stiefel numbers (cf. [[Stiefel number|Stiefel number]]) (see [[#References|[11]]]): If $ M ^ {n - 1 } $ | ||
+ | is a closed equipped manifold and if $ M ^ {n - 1 } = \partial V $, | ||
+ | then the characteristic Stiefel–Whitney number $ w _ {n} ( \nu ) [ V, M] $ | ||
+ | of the normal bundle $ \nu $ | ||
+ | is the same as the Hopf invariant $ H _ {2} ( f ) $ | ||
+ | of the mapping $ f: S ^ {n + r - 1 } \rightarrow S ^ {r} $ | ||
+ | that is a representative of the class of equipped cobordisms of $ M ^ {n - 1 } $. | ||
− | The Adams–Novikov spectral sequence makes it possible to construct higher Hopf invariants. Namely, one defines inductively the invariants | + | The Adams–Novikov spectral sequence makes it possible to construct higher Hopf invariants. Namely, one defines inductively the invariants $ q _ {i} : \mathop{\rm ker} q _ {i - 1 } \rightarrow E _ \infty ^ {i,*} $ |
+ | and $ q _ {0} : \pi _ {*} ^ {S} \rightarrow E _ \infty ^ {0,*} $( | ||
+ | see [[#References|[12]]]). From the form of the differentials of this spectral sequence it follows that | ||
− | + | $$ | |
+ | \mathop{\rm Ext} _ {AU} ^ {i, * } | ||
+ | ( \Omega _ {U} , \Omega _ {U} ) \supset \ | ||
+ | E _ \infty ^ {i, * } ,\ \ | ||
+ | i = 0, 1, 2, 3 | ||
+ | $$ | ||
− | (where | + | (where $ \Omega _ {U} $ |
+ | is the ring of complex point cobordisms); therefore, for $ i = 0, 1, 2, 3 $, | ||
+ | the invariants $ q _ {i} $ | ||
+ | lie in $ \mathop{\rm Ext} _ {AU} ^ {i,*} ( \Omega _ {U} , \Omega _ {U} ) $ | ||
+ | and are called the Hopf–Novikov invariants. For $ i = 1 $ | ||
+ | one obtains the Adams invariant. | ||
− | The values that a Hopf invariant can take are not arbitrary. For example, for a mapping | + | The values that a Hopf invariant can take are not arbitrary. For example, for a mapping $ f: S ^ {4n + 1 } \rightarrow S ^ {2n + 1 } $ |
+ | the Hopf invariant is always 0. The Hopf invariant modulo $ p $, | ||
+ | $ H _ {(} p) : \pi _ {2mp} ( S ^ {2m + 1 } ) \rightarrow \mathbf Z _ {p} $, | ||
+ | is trivial, except when $ p = 2 $, | ||
+ | $ m = 1, 2, 4 $ | ||
+ | and $ p > 2 $, | ||
+ | $ m = 1 $. | ||
+ | On the other hand, for any even number $ k $ | ||
+ | there exists a mapping $ f: S ^ {4n - 1 } \rightarrow S ^ {2n} $ | ||
+ | with Hopf invariant $ k $( | ||
+ | $ n $ | ||
+ | is arbitrary). For $ n = 1, 2, 4 $ | ||
+ | there exists mappings $ f: S ^ {4n - 1 } \rightarrow S ^ {2n} $ | ||
+ | with Hopf invariant 1. | ||
====References==== | ====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> | <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> |
Latest revision as of 22:11, 5 June 2020
An invariant of a homotopy class of mappings of topological spaces. It was first defined by H. Hopf ([1], [2]) for mappings of spheres $ f: S ^ {2n - 1 } \rightarrow S ^ {n} $.
Let $ f: S ^ {2n - 1 } \rightarrow S ^ {n} $ 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 $ S ^ {n} $ and $ S ^ {2n - 1 } $. Then the Hopf invariant is defined as the linking coefficient of the $ ( n - 1) $- dimensional disjoint submanifolds $ f ^ { * } ( a) $ and $ f ^ { * } ( b) $ in $ S ^ {2n - 1 } $ for any distinct $ a, b \in S ^ {n} $.
The mapping $ f: S ^ {2n - 1 } \rightarrow S ^ {n} $ determines an element $ [ f] \in \pi _ {2n - 1 } ( S ^ {n} ) $, and the image of the element $ [ f] $ under the homomorphism
$$ \pi _ {2n - 1 } ( S ^ {n} ) = \ \pi _ {2n - 2 } ( \Omega S ^ {n} ) \rightarrow ^ { h } \ H _ {2n - 2 } ( \Omega S ^ {n} ) = \mathbf Z $$
coincides with the Hopf invariant $ H ( f ) $( here $ h $ is the Hurewicz homomorphism) [3].
Suppose now that $ f: S ^ {2n - 1 } \rightarrow S ^ {n} $ is a mapping of class $ C ^ {2} $ and that a form $ \Omega \in \Lambda ^ {n} S ^ {n} $ is a generator of the integral cohomology group $ H ^ {n} ( S ^ {n} , \mathbf Z ) $. For such a form one may take, for example, $ \Omega = ( dV)/( \mathop{\rm vol} S ^ {n} ) $, where $ dV $ is the volume element on $ S ^ {n} $ in some metric (for example, in the metric given by the imbedding $ S ^ {n} \subset \mathbf R ^ {n + 1 } $), and $ \mathop{\rm vol} S ^ {n} $ is the volume of the sphere $ S ^ {n} $. Then the form $ f ^ { * } ( \Omega ) \in \Lambda ^ {n} S ^ {2n - 1 } $ is closed and it is exact because the group $ H ^ {n} ( S ^ {2n - 1 } , \mathbf Z ) $ is trivial. Thus, $ f ^ { * } ( \Omega ) = d \theta $ for some form $ \theta \in \Lambda ^ {n - 1 } S ^ {2n - 1 } $. A formula for the computation of the Hopf invariant is (see [4]):
$$ H ( f ) = \ \int\limits _ {S ^ {2n - 1 } } \theta \wedge d \theta . $$
The definition of the Hopf invariant can be generalized (see [5], [6]) to the case of mappings $ f: S ^ {m} \rightarrow S ^ {n} $ for $ m \leq 4n - 4 $. In this case there is a decomposition
$$ \tag{* } \pi _ {m} ( S ^ {n} \lor S ^ {n} ) = \ \pi _ {m} ( S ^ {n} ) \oplus \pi _ {m} ( S ^ {n} ) \oplus \pi _ {m} ( S ^ {2n - 1 } ) \oplus \mathop{\rm ker} k _ {*} , $$
where
$$ k _ {*} : \pi _ {m + 1 } ( S ^ {n} \times S ^ {n} ,\ S ^ {n} \lor S ^ {n} ) \rightarrow \ \pi _ {m + 1 } ( S ^ {2n} ) $$
is the homomorphism induced by the projection $ k: ( S ^ {n} \times S ^ {n} , S ^ {n} \lor S ^ {n} ) \rightarrow ( S ^ {n} , \mathop{\rm pt} ) $. Let $ g: S ^ {n} \rightarrow S ^ {n} \lor S ^ {n} $ be the mapping given by contracting the equator of the sphere $ S ^ {n} $ to a point. Then the Hopf invariant is defined as the homomorphism
$$ H: \pi _ {m} ( S ^ {n} ) \rightarrow \pi _ {m} ( S ^ {2n - 1 } ) $$
under which $ [ f] \in \pi _ {m} ( S ^ {n} ) $ is transformed to the projection of the element $ [ g \circ f] \in \pi _ {m} ( S ^ {n} \lor S ^ {n} ) $ onto the direct summand $ \pi _ {m} ( S ^ {2n - 1 } ) $ in the decomposition (*). Since $ \pi _ {2n - 1 } ( S ^ {2n - 1 } ) = \mathbf Z $, for $ m = 2n - 1 $ one obtains the usual Hopf invariant. The generalized Hopf invariant is defined as the composite $ H ^ {*} $ of the homomorphisms
$$ \pi _ {m} ( S ^ {n} ) \rightarrow ^ { {g _ *} } \ \pi _ {m} ( S ^ {n} \lor S ^ {n} ) \rightarrow ^ { p } \ $$
$$ \rightarrow ^ { p } \pi _ { m + 1 } ( S ^ {n} \times S ^ {n} , S ^ {n} \lor S ^ {n} ) \rightarrow ^ { {k _ * } } \pi _ {m + 1 } ( S ^ {2n} ), $$
where $ p $ is the projection of the group $ \pi _ {m} ( S ^ {n} \lor S ^ {n} ) $ onto the direct summand $ \pi _ {m + 1 } ( S ^ {n} \times S ^ {n} , S ^ {n} \lor S ^ {n} ) $, and the homomorphisms $ g _ {*} $ and $ k _ {*} $ are described above. For $ m \leq 4n - 4 $ the Hopf–Whitehead invariant $ H $ and the Hopf–Hilton invariant $ H ^ {*} $ are connected by the relation $ H ^ {*} = S \circ H $, where $ S: \pi _ {m} ( S ^ {2n - 1 } ) \rightarrow \pi _ {m + 1 } ( S ^ {2n} ) $ is the suspension homomorphism (see [6]).
Let $ f: S ^ {2n - 1 } \rightarrow S ^ {n} $ be a mapping and let $ C _ {f} $ be its cylinder (cf. Mapping cylinder). Then the cohomology space $ H ^ {*} ( C _ {f} , S ^ {2n - 1 } ) $ has as homogeneous $ \mathbf Z $- basis a pair $ \{ a, b \} $ with $ \mathop{\rm dim} a = n $ and $ \mathop{\rm dim} b = 2n $. Here the relation $ a ^ {2} = H ( f ) b $ holds (see [7]). If $ n $ is odd, then $ H ( f ) = 0 $( because multiplication in cohomology is skew-commutative).
There is (see [8]) a generalization of the Hopf–Steenrod invariant in terms of a generalized cohomology theory (cf. Generalized cohomology theories). Let $ k $ be the semi-exact homotopy functor in the sense of Dold (see [9]), given on the category of finite CW-complexes and taking values in a certain Abelian category $ A $. Then the mapping of complexes $ f: X \rightarrow Y $ determines an element $ f ^ { * } = d ( f ) \in \mathop{\rm Hom} ( k ( Y), k ( X)) $, where $ \mathop{\rm Hom} $ is the set of morphisms in $ A $. The Hopf–Adams invariant $ e ( f ) $ is defined when $ f ^ { * } = 0 $ and $ d ( Sf ) = 0 $, where $ Sf: SX \rightarrow SY $ is the corresponding suspension mapping. In this case the sequence of cofibrations
$$ X \rightarrow ^ { f } \ Y \rightarrow ^ { f } \ Y \cup _ {f} CX \rightarrow ^ { j } \ SX \mathop \rightarrow \limits ^ {-} Sf \ SY $$
corresponds to an exact sequence in $ A $:
$$ 0 \leftarrow k ( X) \leftarrow ^ { {i _ *} } \ k ( Y \cup _ {f} CX) \leftarrow ^ { {j _ *} } \ k ( SX) \leftarrow 0, $$
which determines the Hopf–Adams–Steenrod invariant $ e ( f ) = \mathop{\rm Ext} ^ {1} ( k ( Y), k ( X)) $.
In the case of the functor $ k = H ^ {*} ( - ; \mathbf Z _ {2} ) $ taking values in the category of modules over the Steenrod algebra modulo 2, one obtains the Hopf–Steenrod invariant $ H _ {2} ( f ) \in \mathbf Z $ of a mapping $ f: S ^ {m} \rightarrow S ^ {n} $ for $ m > n $( see [7]). The cohomology space $ H ^ {*} ( C _ {f} , S ^ {m} ; \mathbf Z _ {2} ) $ has as $ \mathbf Z _ {2} $- basis a pair $ \{ a, b \} $ with $ \mathop{\rm dim} a = n $ and $ \mathop{\rm dim} b = m + 1 $, and then
$$ Sq ^ {m - n + 1 } a = \ H _ {2} ( f ) b. $$
The Hopf invariant $ H _ {p} $ modulo $ p $( where $ p $ is a prime number) is defined as the composite of the mappings
$$ \pi _ {2pn} ( S ^ {2n + 1 } ) _ {(} p) \mathop \rightarrow \limits ^ \approx \ \pi _ {2pn - 2 } ( \Omega ^ {2} S ^ {2n + 1 } ) _ {(} p) \rightarrow $$
$$ \rightarrow \ \pi _ {2pn - 2 } ( \Omega ^ {2} S ^ {2n + 1 } , S ^ {2n - 1 } ) _ {(} p) \rightarrow $$
$$ \rightarrow \ H _ {2pn - 2 } ( \Omega ^ {2} S ^ {2n + 1 } , S ^ {2n - 1 } ) _ {(} p) = \mathbf Z /p, $$
where $ ( X, Y) _ {p} $ is the localization by $ p $ of the pair of spaces (see [10]). Let
$$ S: \pi _ {4n - 1 } ( S ^ {2n} ) \rightarrow \ \pi _ {4n} ( S ^ {2n + 1 } ) $$
be the suspension homomorphism. Then $ H _ {2} ( Sf ) = H _ {2} ( f ) $( see [10]). The Hopf invariant $ H ( f ) $ can also be defined in terms of the Stiefel numbers (cf. Stiefel number) (see [11]): If $ M ^ {n - 1 } $ is a closed equipped manifold and if $ M ^ {n - 1 } = \partial V $, then the characteristic Stiefel–Whitney number $ w _ {n} ( \nu ) [ V, M] $ of the normal bundle $ \nu $ is the same as the Hopf invariant $ H _ {2} ( f ) $ of the mapping $ f: S ^ {n + r - 1 } \rightarrow S ^ {r} $ that is a representative of the class of equipped cobordisms of $ M ^ {n - 1 } $.
The Adams–Novikov spectral sequence makes it possible to construct higher Hopf invariants. Namely, one defines inductively the invariants $ q _ {i} : \mathop{\rm ker} q _ {i - 1 } \rightarrow E _ \infty ^ {i,*} $ and $ q _ {0} : \pi _ {*} ^ {S} \rightarrow E _ \infty ^ {0,*} $( see [12]). From the form of the differentials of this spectral sequence it follows that
$$ \mathop{\rm Ext} _ {AU} ^ {i, * } ( \Omega _ {U} , \Omega _ {U} ) \supset \ E _ \infty ^ {i, * } ,\ \ i = 0, 1, 2, 3 $$
(where $ \Omega _ {U} $ is the ring of complex point cobordisms); therefore, for $ i = 0, 1, 2, 3 $, the invariants $ q _ {i} $ lie in $ \mathop{\rm Ext} _ {AU} ^ {i,*} ( \Omega _ {U} , \Omega _ {U} ) $ and are called the Hopf–Novikov invariants. For $ i = 1 $ one obtains the Adams invariant.
The values that a Hopf invariant can take are not arbitrary. For example, for a mapping $ f: S ^ {4n + 1 } \rightarrow S ^ {2n + 1 } $ the Hopf invariant is always 0. The Hopf invariant modulo $ p $, $ H _ {(} p) : \pi _ {2mp} ( S ^ {2m + 1 } ) \rightarrow \mathbf Z _ {p} $, is trivial, except when $ p = 2 $, $ m = 1, 2, 4 $ and $ p > 2 $, $ m = 1 $. On the other hand, for any even number $ k $ there exists a mapping $ f: S ^ {4n - 1 } \rightarrow S ^ {2n} $ with Hopf invariant $ k $( $ n $ is arbitrary). For $ n = 1, 2, 4 $ there exists mappings $ f: S ^ {4n - 1 } \rightarrow S ^ {2n} $ with Hopf invariant 1.
References
[1] | H. Hopf, "Ueber die Abbildungen der dreidimensionalen Sphäre auf die Kügelfläche" Math. Ann. , 104 (1931) pp. 639–665 |
[2] | H. Hopf, "Ueber die Abbildungen von Sphären niedriger Dimension" Fund. Math. , 25 (1935) pp. 427–440 |
[3] | J.-.P. Serre, "Groupes d'homotopie et classes de groupes abéliens" Ann. of Math. , 58 : 2 (1953) pp. 258–294 |
[4] | J.H.C. Whitehead, "An expression of the Hopf invariant as an integral" Proc. Nat. Acad. Sci. USA , 33 (1937) pp. 117–123 |
[5] | J.H.C. Whitehead, "A generation of the Hopf invariant" Ann. of Math. (2) , 51 (1950) pp. 192–237 |
[6] | P. Hilton, "Suspension theorem and generalized Hopf invariant" Proc. London. Math. Soc. (3) , 1 : 3 (1951) pp. 462–493 |
[7] | N. Steenrod, "Cohomologies invariants of mappings" Ann. of Math. (2) , 50 (1949) pp. 954–988 |
[8] | J. Adams, "On the groups ![]() |
[9] | A. Dold, "Halbexakte Homotopiefunktoren" , Springer (1966) |
[10] | D. Husemoller, "Fibre bundles" , McGraw-Hill (1966) |
[11] | R.E. Stong, "Notes on cobordism theory" , Princeton Univ. Press (1968) |
[12] | 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 |
[13] | J.F. Adams, "On the non-existence of elements of Hopf invariant one" Ann. of Math. , 72 (1960) pp. 20–104 |
Hopf invariant. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hopf_invariant&oldid=47270