― 28 ―The Pontrjagin classes in the BSS complex
Naoya SUZUKI（平成28年11月 1 日受理）
We exhibit cocycles in the Bott-Shulman-Stasheﬀ complex which represent the Pontrjaginclasses.

1.

Introduction

In [1], Dupont introduced a double complex
on a simplicial manifold and showed how to use
it to construct a homomorphism from I ＊ (G), theG-invariant polynomial ring over Lie algebra ,to H ＊ (BG), the cohomology ring of the classifying
space BG. This map is called the Bott-Shulman
map. There he used the property of H ＊ (BG)
that it is isomorphic to the total cohomology ofa complex Ω ＊ (NG (＊)) which is associated to asimplicial manifold {NG (＊)}. The image of theBott-Shulman map in Ω＊ (NG (＊)) is called the BottShulman-Stasheﬀ form. In [3], Shulman gave theshape of it.

－1
φs : ＝ h1 · · · hs －1dhsh－1
s · · · h1 ,∑ sgn(σ) φSp－q ＝In [4], the author exhibited a cocycle inΩ (NG (＊)) which represents the Chern characterin the case of G ＝ GL (n, ).
In this paper, as an application of the resultin [4], we exhibit cocycles which represent thePontrjagin classes.
＊2. The Chern charaters in the BSS complex
In this section we recall the results in [4] that)) which represent
gives cocycles in Ω＊ (NGL (n,the Chern charaters.
we set· · · φσ ( p － q －1) ＋1.

Here h i is the i-th factor of NG (＊). Then thefollowing theorem holds.
Theorem 2.1 ([4]). For (1 ≤ i < j ≤ p－q ＋1), we set :2Rij ＝ (φi ＋ φi ＋1＋ · · · ＋ φj －1)

Then the cocycle in Ω p＋q (NG ( p － q)) (0 ≤ q ≤ p －1)
which represents the p-th Chern character chp is（ ）(－1)
p1
1( p －1)! 2πi( p － q)( p － q －1)/2

×∑ (φ ∧ H (S )trTheorem 1.1 (Shulman). In Ω ＊ (NG (＊)), thecharacteristic classes Φ which are given as the image of
Bott-Shulman map is of the following form:
Φ＝Φ q－1＋Φ q－2 ＋ · · · ＋Φ0q＋i
Φi ∈ Ω (NG ( q－i )).

σ (1) ＋1σ∈ p－q－11∫ ∏(t t )× ∆ p－qi －1 j －1qp－q

dt1 ∧ · · · ∧ dtp－q).

aij(Hq (S))

i<jHere Hq (Sp－q) means the term that Rij (1 ≤ i < j ≤ p－q＋1) are put q -times between φk and φl in Sp－q permitting
overlaps;. aij (Hq (Sp－q)) means the number of Rij in it. ∑means the sum of all such terms.
3.

The Pontrjagin classes in the BSS complex

In this section we take G ＝ GL(n, ) andexhibit cocycles in Ω＊(NG) which represent thePontrjagin classes.
3.1

The first Pontrjagin class in the BSScomplex

The invariant polynomials which give the ﬁrstPontrjagin class and the second Chern character
平成29年２月

― 29 ―The Pontrjagin classes in the BSS complex

are the same.
theorem.

Hence we obtain the following

Theorem 3.1. The cocycle which represents the firstPontrjagin class in Ω4 (NG) is the sum of the following
P1,3 ∈ Ω3 (G) and P2,2 ∈ Ω2 (NG (2)):

（ ）

P1,3＝

1 2 －1tr (h －1 dh)3,2π
6（ ）P2,2＝3.2

1 2 1－1
tr (dh1dh2h－1
2 h1 ).
2π2

The second Pontrjagin class in the BSScomplex

The relation between the invariant polynomials which give the second Pontrjagin class p2 andthe Chern characters is given as: p2＝－6ch4＋ 12 ch22.
Therefore the following theorems hold true.
Theorem 3.2. The cochain P 1,7 in Ω 7 (G) whichcorresponds to the second Pontrjagin class is given as
follows:
P1,7＝ ( －6)

（ ）
14!

1 2 1tr (h －1 dh)7.
2π 35

Theorem 3.3. The cochain P2,6 in Ω6 (NG (2)) whichcorresponds to the second Pontrjagin class is given as
follows:
P2,6＝（ ）1 2－1
P 1,3＋ (－6)
23!5!

12π4Theorem 3.4. The cochain P3,5 in Ω5 (NG (3)) whichcorresponds to the second Pontrjagin class is given as
follows:（ ）1－1(P1,3 P2,2＋P2,2 P1,3)＋(－6)
2
3!5!

秋田高専研究紀要第52号

Theorem 3.5. The cochain P4,4 in Ω4 (NG (4)) whichcorresponds to the second Pontrjagin class is given as
follows:
P4,4＝－1 －1－1－1 －1×tr(3dh1dh2h－12 dh2h2 h1 dh1h1 dh1dh2h2 h1－1－1－1－1 －1＋3dh1dh2h－1
2 h1 dh1h1 dh1dh2h2 dh2h2 h12 －1 －1－1－1－1＋6dh1h－1
1 dh1h1 dh1dh2h2 dh2h2 dh h2 h1－1－1－1－1－1＋6dh1h1 dh1h1 dh1h1 dh1h1 dh1dh2h－1
2 h1
－1
－1 －1
－1 －1＋6dh1h－1
1 dh1h1 dh1dh2h2 h1 dh1dh2h2 h1－1－1－1－1 －1＋6dh1dh2h－1
2 h1 dh1dh2h2 dh2h2 dh2h2 h1－1－1－1－1－1＋6dh1dh2h2 dh2h2 dh2h2 dh2h2 dh2h－1
2 h1
－1
－1 －1
－1 －1＋2dh1dh2h－1
2 h1 dh1dh2h2 h1 dh1dh2h2 h1 ).

P3,5＝－1－1 －1 －1×tr(6dh1h－1
1 dh1h1 dh1dh2dh3h3 h2 h1－1－1 －1－1 －1－6dh1h－1
1 dh1h1 dh1h2dh3h3 h2 dh2h2 h1－1－1－1－1 －1－6dh1h2dh3h－1
3 h2 dh2h2 dh2h2 dh2h2 h1
－1－1
－1 －1
－1－6dh1h2dh3h3 dh3h3 dh3h3 h2 dh2h－1
2 h1－1－1 －1 －1＋6dh1dh2h－1
2 h1 dh1dh2dh3h3 h2 h1－1－1 －1－1 －1－6dh1dh2h－1
2 h1 dh1h2dh3h3 h2 dh2h2 h1－1 －1－1 －1 －1＋6dh1dh2dh3h3 h2 dh2dh3h3 h2 h1－1－1 －1－1 －1－6dh1h2dh3h－1
3 h2 dh2dh3h3 h2 dh2h2 h1－1 －1－1－1 －1 －1＋3dh1dh2h2 h1 dh1h1 dh1h2dh3h3 h2 h1－1 －1－1－1 －1－3dh1h2dh3h－1
3 h2 h1 dh1h1 dh1dh2h2 h1－1 －1－1 －1 －1＋3dh1h2dh3h－1
3 dh3h3 h2 dh2dh3h3 h2 h1－1 －1－1－1 －1－3dh1h2dh3h3 h2 dh2dh3h3 dh3h－1
3 h2 h1－1 －1－1 －1 －1＋3dh1dh2dh3h－1
3 h2 h1 dh1h2dh3h3 h2 h1－1－1 －1－3dh1h2dh3h－1
3 h2 dh2h2 h1－1 －1 －1
dh1h2dh3h3 h2 h1－1－1 －1 －1＋8dh1dh2h－1
2 dh2h2 dh2dh3h3 h2 h1－1－1 －1－1－4dh1dh2h2 dh2dh3h3 h2 dh2h－1
2 h1－1－1 －1 －1＋4dh1dh2dh3h－1
3 dh3h3 dh3h3 h2 h1－1－1－1 －1＋4dh1dh2dh3h－1
3 h2 dh2h2 dh2h2 h1－1－1－1 －1 －1＋2dh1h1 dh1dh2h2 dh2dh3h3 h2 h1－1－1－1 －1－1－2dh1h2dh3h－1
3 h2 dh2h2 dh2h2 h1 dh1h1－1－1 －1 －1＋2dh1dh2h－1
2 dh2dh3h3 dh3h3 h2 h1－1－1－1 －1－1 －1－2dh1h2 dh3h3 dh3h3 h2 dh2h－1
2 dh2h2 h1－1－1 －1 －1＋ dh1h－1
1 dh1dh2dh3h3 dh3h3 h2 h1－1－1－1 －1－1－ dh1h1 dh1h2dh3h3 dh3h3 h2 dh2h－12 h1 ).

1 42π（ ）1 2
1P 2,2＋ ( －6)
24!3!

12π4

－1 －1 －1×tr(dh1dh2dh3dh4h－1
4 h3 h2 h1－1－1 －1 －1
－ dh1dh2dh3dh4h－1
4 h3 dh3h3 h2 h1－1－1 －1 －1 －1－ dh1h2dh3h－1
3 h2 dh2h3dh4h4 h3 h2 h1－1 －1 －1
－1＋ dh1h2dh3dh4h4 h3 h2 dh2h－1
2 h1－1－1 －1－1 －1－ dh1h2h3dh4h－1
4 h3 dh3h3 h2 dh2h2 h1－1 －1－1 －1 －1＋ dh1h2h3dh4h－1
4 h3 h2 dh2dh3h3 h2 h1 ).

Errata 1. There are some typos in [5]. In Theorem
3.2 and Theorem 4.1, the word “to” after “represents”
should be omitted. In Theorem 3.2, “folllows” should be
replaced with “follows”.

― 30 ―
Naoya SUZUKI

References[1] J. L. Dupont, Simplicial de Rham cohomology
and characteristic classes of ﬂat bundles, Top.
Vol 15 (1976), 233-245, Perg Press.
[2] J.L. Dupont, Curvature and CharacteristicClasses, Lecture Notes in Math. 640, Springer
Verlag, (1978).
[3] H. Shulman, Characteristic Classes andFoliations, Ph.D.Thesis, University ofCalifornia, Berkeley, 1972.
[4] N. Suzuki, T h e C h e r n c h a r a c t e r i n t h e
Simplicial de Rham Complex. Nihonkai
Mathematical Journal, Vol.26, No1, pp.1-13,(2015).
[5] N. Suzuki, On some characteristic classes of
the universal torus bundle in the BSS complex,
秋田高専研究紀要 第51号，pp.73-75，（平成28年）.

平成29年２月