On homological stability for orthogonal groups and special orthogonal groups

📝 Abstract

The problem of homological stability helps us to catch the structure of group homology. We calculate homological stability of special orthogonal groups, and we also calculate the stability of orthogonal groups with determinant-twisted coefficients under a certain good situation. We also get some results about the structure of these homology.

💡 Analysis

The problem of homological stability helps us to catch the structure of group homology. We calculate homological stability of special orthogonal groups, and we also calculate the stability of orthogonal groups with determinant-twisted coefficients under a certain good situation. We also get some results about the structure of these homology.

📄 Content

arXiv:1106.2883v1 [math.KT] 15 Jun 2011 ON HOMOLOGICAL STABILITY FOR ORTHOGONAL GROUPS AND SPECIAL ORTHOGONAL GROUPS MASAYUKI NAKADA Abstract. The problem of homological stability helps us to catch the struc- ture of group homology. We calculate homological stability of special orthog- onal groups, and we also calculate the stability of orthogonal groups with determinant-twisted coefficients under a certain good situation. We also get some results about the structure of these homology. Contents 1. Introduction 1 1.1. Notations and Preliminaries 1 1.2. Results 2 2. Involution σ 2 3. Induction algorithm 4 4. On structures of σ-coinvariant part 6 5. Proof of theorem 1.3 7 6. Applications 10 6.1. Proof of theorem 1.5 10 6.2. Some applications 11 References 12

1. Introduction 1.1. Notations and Preliminaries. In this paper, F is an infinite Pythagorean field of char(F) ̸= 2. A field F is Pythagorean if the sum of any two squares is a square. Other than algebraically closed fields, the real numbers R is a typical example. Let q(x) = Pn i=1 x2 i be the Euclidean quadratic form on F n. We denote by On = On(F, q) the corresponding orthogonal group and by SOn = SOn(F, q) the corresponding special orthogonal group of degree n. We denote by S = S(F n) the unit sphere {x ∈F n; q(x) = 1}. We write by Zt the determinant-twisted On-module which admits the twisted action by the determinant. We consider, for any integer n ≥0, that F n is isometrically embedded in F n+1 as x 7→(0, x). This defines an inclusion map (1.1) inc: On →On+1, A 7→  1 0 0 A  for n ≥1 and, in case n = 0, we consider O0 as the trivial group. The inclusion (1.1) induces the map of homology groups (1.2) Hi(inc): Hi(On) −→Hi(On+1) Date: May 25, 2022. 2010 Mathematics Subject Classification. 20J05. Key words and phrases. Group homology, Homological stability, Scissors congruence. 1 2 MASAYUKI NAKADA in the i-th degree. We understand the coefficient of homology is Z if it is omitted. It is well known that every isomorphic embedding induces an inclusion inc: On ֒→ On+1 which are conjugate each other by the theorem of Witt and induces the same map in homology. In the same way, from the inclusion inc: SOn ֒→SOn+1 we have Hi(inc): Hi(SOn) →Hi(SOn+1). Hi(SOn) admits an involution induced from the short exact sequence (1.3) 1 / SOn i / On det / Z/2 / 1. Here Z/2 means the multiplicative group {±1}, i is the natural inclusion and det is the determinant homomorphism. We denote by σ this involution. 1.2. Results. The problem of homological stability of On was first studied by Sah in [5] in case F = R. Cathelineau generalised the result of Sah for any infinite Pythagorean fields in [2]. In [2], Cathelineau proved the following; let On = On(F, q) be the orthogonal group over an infinite Pythagorean field F with Euclidean quadratic form q. Proposition 1.1 ( [2,5]). The map Hi(inc): Hi(On) →Hi(On+1) is bijective for i < n, and surjective for i ≤n. In Sah’s paper [5] only H2 of the stability for SOn(R) was studied. In [2] Cathe- lineau proved the following result for SOn = SOn(F, q): Proposition 1.2 ( [2]). The map Hi(inc): Hi(SOn, Z[1/2]) →Hi(SOn+1, Z[1/2]) is bijective for 2i < n, and surjective for 2i ≤n. In case F is quadratically closed, it is known that the obstruction to stability for SOn with coefficient Z[1/2] is the Milnor K group KM n (F) [2]. We make precise the result of Cathelineau’s result (proposition 1.2). Our result for special orthogonal groups is the following: Theorem 1.3. The map Hi(inc): Hi(SOn) →Hi(SOn+1) is bijective for 2i < n, and surjective for 2i ≤n. In the proof of theorem 1.3 we also get the following corollary (see section 5). Corollary 1.4. For 2i < n, the group Hi(SOn) is isomorphic to its own σ-inariant part. Another implication is; Theorem 1.5. The map Hi(inc, Zt): Hi(On, Zt) →Hi(On+1, Zt) is bijective for 2i < n, and surjective for 2i ≤n. The group Hn(On, Zt) plays an important role in the problem of spherical scissors congruence (see [3]). Acknowledgements. The author would like to thank Masana Harada for his helpful supports.
2. Involution σ In this section we study the involution on Hi(SOn) induced from the extension (1.3). Let R be a commutative ring, G be a group and M be a left RG-module. Let γ be an element in G. We can define an endomorphism γ∗on Hi(G, M) by γ∗([g1| . . . |gi] ⊗m) = [γg1γ−1| . . . |γgiγ−1] ⊗γm ON HOMOLOGICAL STABILITY FOR On AND SOn 3 using the standard bar resolution. Here g1, . . . , gi ∈G and m ∈M. Notice that (2.1) the endomorphism γ∗is chain homotopic to the identity. For the proof, see [3, Lemma 5.4] for instance. The sequence (1.3) splits by ι: Z/2 →On, ±1 7→  ±1 0 0 1n−1  , where 1n−1 means the (n −1)-unit matrix. The sequence (1.3) induces the action of Z/2 = {±1} on SOn. The (−1)-action −1·g := −1 0 0 1n−1  g −1 0 0 1n−1 −1 defines an involution σ on Hi(SOn). Observe that the following diagram (2.2) Hi(SOn) σ  Hi(inc)/ Hi(SOn+1) σ  Hi(SOn) Hi(inc) / Hi(SOn+1) is commutative, for inc(σ·g) =   1 0 0 −1 0 0

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