Construction of Homotopic Invariants of Maps from Spheres to Compact Closed Manifolds
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Journal of Mathematical Sciences, Vol. 249, No. 6, September, 2020
CONSTRUCTION OF HOMOTOPIC INVARIANTS OF MAPS FROM SPHERES TO COMPACT CLOSED MANIFOLDS I. S. Zubov State Socio-Humanitarian University 30, Zelenaya St., Kolomna 140410, Russia reestr [email protected]
UDC 517.925 + 512.77
We study the homotopic classifications of maps from circles and spheres to manifolds and compare the classical approach to define the Hopf invariant with the approach based on Chen’s iterated integrals. Bibliography: 5 titles.
1
Iterated Integrals
Let ω1 , ω2 , . . . , ωr be differential 1-forms on a smooth manifold M . Denote by Px0 (M ) the space of paths γ : I = [0, 1]→ M with starting point x0 = γ(0). By an iterated integral of ω1 , ω2 , · · · ωr we mean a function
ω1 . . . ωr : Px0 (M ) → R on the path space Px0 (M ) such that for a path γ
it is inductively defined by the equality
ω1 . . . ωr−1 ωr ,
ω1 . . . ωr = γ
γ
γ τ = γ(τ t),
t ∈ [0, 1].
γτ
ω1 is a usual curvilinear integral. For complex-valued differential 1-forms
We note that γ
ω1 . . . ωr the iterated integrals take the values in C. We denote by Pxx01 (M ). the space of paths ω1 . . . ωr γ such that γ(0) = x0 and γ(1) = x1 and by Bs (M ) the space of iterated integrals of length r s. We consider some properties of iterated integrals.
γ
1.1. The product of iterated integrals is a linear combination of iterated integrals.
Translated from Problemy Matematicheskogo Analiza 103, 2020, pp. 101-112. c 2020 Springer Science+Business Media, LLC 1072-3374/20/2496-0929
929
Theorem 1.1. The product of iterated integrals of length k and l respectively is equal to the sum of iterated integrals of length k + l ω1 · · · ωk · ωk+1 · · · ωk+l = ωσ(1) · · · ωσ(k+l) , γ
σ∈Sk,l γ
γ
where the sum is taken over all shuffles (k, l) in the permutation group Sn+k . Proof. Let γ ∗ ωi = fi (t)dt, i = 1, . . . , k + l. By the definition of iterated integrals, ∗ ∗ ω1 · · · ωk · ωk+1 · · · ωk+l = γ ω1 · · · γ ωk · γ ∗ ωk+1 · · · γ ∗ ωk+l γ
γ
=
k Ck
[0,1]
fi (ti )dt1 · · · dtk ·
i=1
Cl
k+l
[0,1]
k+l
fi (ti )dtk+1 · · · dtk+l =
i=k+1
Ck ×Cl =
∪
σ∈Sk,l
σ(Ck+l )
fi (ti )dt1 · · · dtk+l ,
i=1
where the last equality holds in view of the Fubini theorem. Since the integral over the union of measurable sets is equal to the sum of integrals over each set in this union provided that the intersection of these sets has zero measure, we have Ck ×Cl =
=
∪
k+l
σ∈Sk,l
σ(Ck+l )
fi (ti )dt1 · · · dtk+l =
i=1
σ∈Sk,lσ(C
k+l )
k+l
fi (ti )dt1 · · · dtk+l
i=1
ωσ−1 (1) · · · ωσ−1 (k+l) ,
σ∈Sk,l γ
where Cn is an n-dimensional standard simplex. 1.2. We consider differential forms ω1 , . . . , ωr on M n and a path γ : [0, 1] → M n . We set = γ(t(τ )), where t(τ ) : [0, 1] → [0, 1] is a change of variable. If t(τ ) is monotonically increasing, then the equivalence class of paths, up to a change of variable, is called an oriented curve. We use the induction on r to prove the invariance property under a differentiable mo
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