PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 101(115) (2017), 1–10 https://doi.org/10.2298/PIM1715001P

NEW IMMERSION THEOREMS FOR GRASSMANN MANIFOLDS G_{3,n}

Zoran Z. Petrović and Branislav I. Prvulović

Abstract. A Gröbner basis for the ideal determining mod 2 cohomology of Grassmannian G_{3,n} , obtained in a previous paper, is used, along with the method of obstruction theory, to establish some new immersion results for these manifolds.

1. Introduction

The theory of Gröbner bases is one of the most powerful tools for deciding whether a certain polynomial in two or more variables belongs to a given ideal. An example where this problem is of particular interest is the mod 2 cohomology algebra of Grassmann manifold G_{k,n} = O(n + k)/O(n) × O(k). By Borel’s description, this algebra is just the polynomial algebra on the Stiefel–Whitney classes w_{1}, w_{2}, …, w_{k} of the canonical vector bundle γ_{k} over G_{k,n} modulo the ideal J_{k,n} generated by the dual classes w̄_{n+1} , w̄_{n+2} , …, w̄_{n+k} .

A reduced Gröbner basis for the ideal J_{2,n} has been obtained in [6]. Based on that result for odd n, some new immersions of Grassmannians G_{2,2l+1} were established.

In [9] reduced Gröbner bases for all ideals J_{k,n} were determined. We use this result and the method of modified Postnikov towers (MPT) developed by Gitler and Mahowald [3] to get new immersion results. In the following, imm(G_{3,n}) stands for the immersion dimension of Grassmannians G_{3,n}

imm(G_{3,n}) := min{d | G_{3,n} immerses into ℝ^{d}}.

Some lower bounds for imm(G_{3,n}) were established by Oproiu in [5], where he used the method of the Stiefel–Whitney classes, and from the general result of Cohen [1] one has an upper bound for imm(G_{3,n}). In [7] it is shown that imm(G_{3,n}) = 6n − 3 if n is a power of two.

Now we have the following new immersion results.

2010 Mathematics Subject Classification: Primary 57R42, 55S45, 55R40; Secondary 13P10.

Key words and phrases: Grassmannians, immersions, Gröbner bases, modified Postnikov towers.

Communicated by Rade Živaljević.

Theorem 1.1. If n ⩾ 3 and n ≡ 1 (mod 8), then G_{3,n} immerses into ℝ^{6n−6} .

This theorem improves Cohen’s result whenever α(3n) < 6 (as usual, α(3n) denotes the number of ones in the binary expansion of 3n). For example, if n = 1 + 2^{r} + ∑ _{j=1}^{s} 2^{r+2j−1} = 1 + 2^{r} + 2^{r+1} · \frac{2^{2s} −1}{3} for some r ⩾ 3 and s ⩾ 0, we have that 3n = 3 + 2^{r} + 2^{r+2s+1} , so α(3n) = 4. When s = 0, i.e., n = 2^{r} + 1 (r ⩾ 3), by Theorem 1.1 and Oproiu’s result, we have that 6 · 2^{r} − 3 ⩽ imm(G_{3,2^{r} +1} ) ⩽ 6 · 2^{r}.

Theorem 1.2. If n ≡ 6 (mod 8), then G_{3,n} immerses into ℝ^{6n−5} .

The best improvement of Cohen’s general result obtained from Theorem 1.2 is in the case n = 2 + ∑ _{j=1}^{s} 2^{2j} , s ⩾ 1. Then 3n = 2 + 2^{2s+2} and so we are able to decrease the upper bound for imm(G_{3,n}) by 3. For example, by this theorem and Oproiu’s result, we have that 29 ⩽ imm(G_{3,6}) ⩽ 31.

Theorem 1.3. If n ⩾ 3 and n ≡ 2 (mod 8), then G_{3,n} immerses into ℝ^{6n−7} .

Again, there is a number of cases in which Theorem 1.3 improves previously known results. In particular, when n = 2^{r} + 2, r ⩾ 3, we have an improvement by 3. In this case, using Oproiu’s result and this theorem, we have 6 · 2^{r} − 3 ⩽ imm(G_{3,2^{r} +2} ) ⩽ 6 · 2^{r} + 5.

In this paper we present only a proof of Theorem 1.1. The other theorems may be proved by using the same techniques.

Remark 1.1. The detailed proofs of all three theorems can be found in [8]. Actually, this paper is an abridged version of [8]. In addition to these proofs, the preprint [8] contains the already mentioned result from [7] and the construction of Gröbner bases for J_{3,n}. This construction is not included in this paper, since these Gröbner bases are obtained in full generality in [9].

2. Gröbner bases

Throughout this section, we denote by ℕ_{0} the set of all nonnegative integers and the set of all positive integers is denoted by ℕ.

Let G_{k,n} be the Grassmann manifold of unoriented k-dimensional vector subspaces in ℝ^{n+k} , and w_{1}, w_{2}, …, w_{k} the Stiefel–Whitney classes of the canonical bundle γ_{k} over G_{k,n} . It is known that the cohomology algebra H^{∗}(G_{k,n} ; ℤ_{2}) is isomorphic to the quotient ℤ_{2}[w_{1}, w_{2}, …, w_{k}]/J_{k,n} of the polynomial algebra ℤ_{2}[w_{1}, w_{2}, …, w_{k}] by the ideal J_{k,n} generated by the polynomials w̄_{n+1} , …, w̄_{n+k} . These are obtained from the equation

(1 + w_{1} + w_{2} + ⋯+ w_{k})(1 + w̄_{1} + w̄_{2} + ⋯) = 1.

For k = 3 (which is the case from now on), one has

w̄_{r} = ∑_{a+2b+3c=r} \binom{a+b+c}{a}\binom{b+c}{b}w_{1}^{a}w_{2}^{b}w_{3}^{c}, r ∈ ℕ,

where it is understood that a, b, c ∈ ℕ_{0}.

Let n ⩾ 3 be a fixed integer. We define a set of polynomials in ℤ_{2}[w_{1}, w_{2}, w_{3}].

NEW IMMERSION THEOREMS FOR GRASSMANN MANIFOLDS G_{3,n} 3

Definition 2.1. For m, l ∈ ℕ_{0}, let

g_{m,l} := ∑_{a+2b+3c=n+1+m+2l}  \binom{a+b+c−m−l}{a}\binom{b+c−l}{b}w_{1}^{a}w_{2}^{b}w_{3}^{c}.

As before, it is understood that a, b, c ∈ ℕ_{0}. Note that g_{0,0} = w̄_{n+1} . The following theorem is a special case of Theorem 14 from [9].

Theorem 2.1. The set G = {g_{m,l} | m + l ⩽ n + 1, m, l ∈ ℕ_{0}} is the reduced Gröbner basis for the ideal J_{3,n} = (w̄_{n+1} , w̄_{n+2} , w̄_{n+3} ) with respect to the grlex ordering on monomials with w_{1} > w_{2} > w_{3}.

The polynomials g_{m,l} ∈ G for l = n + 1 and l = n are calculated directly from Definition 2.1:

g_{0,n+1} = w_{3}^{n+1} , g_{0,n} = w_{1}w_{3}^{n}, g_{1,n} = w_{2}w_{3}^{n}.

Now, the following equalities may be obtained by using the well-known formula \binom{a}{b} = \binom{a−1}{b} + \binom{a−1}{b−1}, a, b ∈ ℤ (it is understood that \binom{a}{b} = 0 if b is negative), along with some convenient index shifting

g_{m+2,l} = g_{m,l+1} + w_{2}g_{m,l} + w_{1}g_{m+1,l} ,
g_{m+1,l+1} = w_{3}g_{m,l} + w_{1}g_{m,l+1} ,
g_{m,l+2} = w_{3}g_{m+1,l} + w_{2}g_{m,l+1} .

We list a few elements from G which will be needed in the following section. One may obtain them by the repeated use of the previous equalities

g_{1,n−2} = w_{1}^{2}w_{2}w_{3}^{n−2} + w_{1}w_{3}^{n−1} + w_{2}^{2}w_{3}^{n−2} ,
g_{2,n−3} = w_{1}^{2}w_{2}^{2}w_{3}^{n−3} + w_{2}^{3}w_{3}^{n−3} + w_{3}^{n−1} ,
g_{3,n−3} = w_{1}w_{2}^{3}w_{3}^{n−3} + w_{2}^{2}w_{3}^{n−2} ,
g_{3,n−4} = w_{1}^{2}w_{2}^{3}w_{3}^{n−4} + w_{1}w_{2}^{2}w_{3}^{n−3} + w_{2}^{4}w_{3}^{n−4} + w_{2}w_{3}^{n−2} ,
g_{5,n−4} = w_{2}^{5}w_{3}^{n−4} + w_{1}w_{3}^{n−1} ,
g_{4,n−5} = w_{1}^{2}w_{2}^{4}w_{3}^{n−5} + w_{2}^{5}w_{3}^{n−5} ,
g_{6,n−5} = w_{2}^{6}w_{3}^{n−5} + w_{3}^{n−1} .

In the following, we will also use the fact that the set {w_{1}^{a}w_{2}^{b}w_{3}^{c} | a + b + c ⩽ n} is a vector space basis for H^{∗}(G_{3,n}; ℤ_{2}) (see, e.g., [9, Proposition 13]).

3. Immersions

As before, let w_{i} be the i-th Stiefel–Whitney class of the canonical vector bundle γ_{3} over G_{3,n} (n ⩾ 3) and let r be the (unique) integer such that 2^{r+1} < 3n < 2^{r+2} , i.e., \frac{2}{3} · 2^{r} < n < \frac{4}{3} · 2^{r}. It is well known (see [5, p. 183]) that for the stable normal bundle ν of G_{3,n} one has

(3.1)  w(ν) = (1 + w_{1}^{4} + w_{2}^{2} + w_{1}^{2}w_{2}^{2} + w_{3}^{2})(1 + w_{1} + w_{2} + w_{3})^{2^{r+1} −n−3} .

For n ⩽ 2^{r} − 3, by the result of Stong [10] ht(w_{1}) = 2^{r} − 1 and by the result of Dutta and Khare [2] ht(w_{2}) ⩽ 2^{r} − 1. Also, w_{3}^{2^{r}} = 0 since 3 · 2^{r} > 3 · (2^{r} − 3) ⩾ 3n = dim(G_{3,n}) and we have that (1 + w_{1} + w_{2} + w_{3})^{2^{r}} = 1. This means that in this case (\frac{2}{3} · 2^{r} < n ⩽ 2^{r} − 3) formula (3.1) simplifies to

(3.2)  w(ν) = (1 + w_{1}^{4} + w_{2}^{2} + w_{1}^{2}w_{2}^{2} + w_{3}^{2})(1 + w_{1} + w_{2} + w_{3})^{2^{r} −n−3} .

In order to shorten the upcoming calculations, we give two equalities concerning the action of the Steenrod algebra A_{2} on H^{∗}(G_{3,n}; ℤ_{2}) which can be obtained by using the basic properties of A_{2} and formulas of Wu and Cartan. It is understood that a, b and c are nonnegative integers.

Sq^{1}(w_{1}^{a}w_{2}^{b}w_{3}^{c}) = (a + b + c)w_{1}^{a+1} w_{2}^{b}w_{3}^{c} + bw_{1}^{a}w_{2}^{b−1} w_{3}^{c+1} ,
Sq^{2}(w_{1}^{a}w_{2}^{b}w_{3}^{c}) = \binom{a+b+c}{2}w_{1}^{a+2} w_{2}^{b}w_{3}^{c} + b(a + c)w_{1}^{a+1} w_{2}^{b−1} w_{3}^{c+1}
+ (b + c)w_{1}^{a}w_{2}^{b+1} w_{3}^{c} + \binom{b}{2}w_{1}^{a}w_{2}^{b−2} w_{3}^{c+2} .

In the rest of the paper, it is understood that n is a fixed integer such that n ⩾ 3 and n ≡ 1 (mod 8).

Lemma 3.1. If ν is the stable normal bundle of G_{3,n}, then

(a) w_{i}(ν) = 0 for i ⩾ 3n − 8;

(b) w_{1}(ν) = w_{2}(ν) = 0;

(c) w_{4}(ν) = w_{2}^{2}.

Proof. As above, let r ⩾ 3 be the integer such that 2^{r+1} < 3n < 2^{r+2} . If n ⩾ 2^{r}, then n must be ⩾ 2^{r} + 1. So we have that 2^{r+1} ⩽ 2n − 2. The top class in expression (3.1), (w_{1}^{2}w_{2}^{2} + w_{3}^{2})w_{3}^{2^{r+1} −n−3} , is in degree 6 + 3 · (2^{r+1} − n − 3) ⩽ 6 + 3 · (n − 5) = 3n − 9 and (a) follows in this case.

If n < 2^{r}, then we actually have that n < 2^{r} − 2 (since n ≡ 1 (mod 8)), so formula (3.2) holds. The top class there is in degree 6 + 3 · (2^{r} − n − 3) and, since 3n > 2^{r+1} , we have that 2^{r} < \frac{3}{2}n, implying 6 + 3 · (2^{r} − n − 3) < 6 + 3 · \frac{n−6}{2} < 6 + 3 · (n − 6) = 3n − 12. This proves (a).

Parts (b) and (c) we read off from formula (3.1) (using the fact that 2^{r+1} − n − 3 ≡ 4 (mod 8))

w_{1}(ν) = (2^{r+1} − n − 3)w_{1} = 0,
w_{2}(ν) = \binom{2^{r+1} −n−3}{2}w_{1}^{2} + (2^{r+1} − n − 3)w_{2} = 0,
w_{4}(ν) = w_{1}^{4} + w_{2}^{2} + \binom{2^{r+1} −n−3}{4}w_{1}^{4} + \binom{2^{r+1} −n−3}{3}\binom{3}{1}w_{1}^{2}w_{2}
+ \binom{2^{r+1} −n−3}{2}\binom{2}{1}w_{1}w_{3} + \binom{2^{r+1} −n−3}{2}w_{2}^{2} = w_{2}^{2},

and the lemma follows. □

Lemma 3.2. For the map Sq^{2} : H^{3n−6} (G_{3,n}; ℤ_{2}) → H^{3n−4} (G_{3,n}; ℤ_{2}), we have

Sq^{2}(w_{1}^{2}w_{2}^{2}w_{3}^{n−4} ) = w_{1}^{2}w_{3}^{n−2} + w_{1}w_{2}^{2}w_{3}^{n−3} + w_{2}^{4}w_{3}^{n−4} + w_{2}w_{3}^{n−2} ;
Sq^{2}(w_{1}w_{2}w_{3}^{n−3} ) = w_{1}^{2}w_{3}^{n−2} + w_{1}w_{2}^{2}w_{3}^{n−3} ;
Sq^{2}(w_{3}^{n−2} ) = w_{1}^{2}w_{3}^{n−2} + w_{2}w_{3}^{n−2} .

NEW IMMERSION THEOREMS FOR GRASSMANN MANIFOLDS G_{3,n} 5

Proof. We use the Gröbner basis G to calculate:

Sq^{2}(w_{1}^{2}w_{2}^{2}w_{3}^{n−4} ) = \binom{n}{2}w_{1}^{4}w_{2}^{2}w_{3}^{n−4} + 2(n − 2)w_{1}^{3}w_{2}w_{3}^{n−3}
+ (n − 2)w_{1}^{2}w_{2}^{3}w_{3}^{n−4} + \binom{2}{2}w_{1}^{2}w_{3}^{n−2}
= w_{1}^{2}w_{2}^{3}w_{3}^{n−4} + w_{1}^{2}w_{3}^{n−2}
= g_{3,n−4} + w_{1}w_{2}^{2}w_{3}^{n−3} + w_{2}^{4}w_{3}^{n−4} + w_{2}w_{3}^{n−2} + w_{1}^{2}w_{3}^{n−2} .

Since g_{m,l} = 0 in H^{∗}(G_{3,n}; ℤ_{2}), we obtain the first equality. Also,

Sq^{2}(w_{1}w_{2}w_{3}^{n−3} ) = \binom{n−1}{2}w_{1}^{3}w_{2}w_{3}^{n−3} + (n − 2)w_{1}^{2}w_{3}^{n−2} + (n − 2)w_{1}w_{2}^{2}w_{3}^{n−3} and using the congruence n ≡ 1 (mod 8), we directly get the second equality. Similarly,

Sq^{2}(w_{3}^{n−2} ) = \binom{n−2}{2}w_{1}^{2}w_{3}^{n−2} + (n − 2)w_{2}w_{3}^{n−2} = w_{1}^{2}w_{3}^{n−2} + w_{2}w_{3}^{n−2} , and we are done. □

Lemma 3.3. The map Sq^{2} : H^{3n−4} (G_{3,n}; ℤ_{2}) → H^{3n−2} (G_{3,n}; ℤ_{2}) is given by the following equalities:

Sq^{2}(w_{1}^{2}w_{3}^{n−2} ) = w_{1}w_{3}^{n−1} + w_{2}^{2}w_{3}^{n−2} ,
Sq^{2}(w_{1}w_{2}^{2}w_{3}^{n−3} ) = Sq^{2}(w_{2}^{4}w_{3}^{n−4} ) = Sq^{2}(w_{2}w_{3}^{n−2} ) = w_{1}w_{3}^{n−1} .

Proof. The set {w_{1}^{2}w_{3}^{n−2} , w_{1}w_{2}^{2}w_{3}^{n−3} , w_{2}^{4}w_{3}^{n−4} , w_{2}w_{3}^{n−2} } is a vector space basis for H^{3n−4} (G_{3,n}; ℤ_{2}). We proceed to the calculation.

Sq^{2}(w_{1}^{2}w_{3}^{n−2} ) = \binom{n}{2}w_{1}^{4}w_{3}^{n−2} + (n − 2)w_{1}^{2}w_{2}w_{3}^{n−2} = w_{1}^{2}w_{2}w_{3}^{n−2}
= g_{1,n−2} + w_{1}w_{3}^{n−1} + w_{2}^{2}w_{3}^{n−2} = w_{1}w_{3}^{n−1} + w_{2}^{2}w_{3}^{n−2} ,
Sq^{2}(w_{1}w_{2}^{2}w_{3}^{n−3} ) = \binom{n}{2}w_{1}^{3}w_{2}^{2}w_{3}^{n−3} + 2(n − 2)w_{1}^{2}w_{2}w_{3}^{n−2}
+ (n − 1)w_{1}w_{2}^{3}w_{3}^{n−3} + \binom{2}{2}w_{1}w_{3}^{n−1} = w_{1}w_{3}^{n−1} ,
Sq^{2}(w_{2}^{4}w_{3}^{n−4} ) = \binom{n}{2}w_{1}^{2}w_{2}^{4}w_{3}^{n−4} + 4 · (n − 4)w_{1}w_{2}^{3}w_{3}^{n−3} + nw_{2}^{5}w_{3}^{n−4}
+ \binom{4}{2}w_{2}^{2}w_{3}^{n−2} = w_{2}^{5}w_{3}^{n−4} = g_{5,n−4} + w_{1}w_{3}^{n−1} = w_{1}w_{3}^{n−1} ,
Sq^{2}(w_{2}w_{3}^{n−2} ) = \binom{n−1}{2}w_{1}^{2}w_{2}w_{3}^{n−2} + (n − 2)w_{1}w_{3}^{n−1} + (n − 1)w_{2}^{2}w_{3}^{n−2}
= w_{1}w_{3}^{n−1} .  □

Lemma 3.4. The map Sq^{1} : H^{3n−3} (G_{3,n}; ℤ_{2}) → H^{3n−2} (G_{3,n}; ℤ_{2}) is given by

Sq^{1}(w_{1}w_{2}w_{3}^{n−2} ) = w_{2}^{2}w_{3}^{n−2} , Sq^{1}(w_{2}^{3}w_{3}^{n−3} ) = Sq^{1}(w_{3}^{n−1} ) = 0.

Proof. We know that the classes w_{1}w_{2}w_{3}^{n−2} , w_{2}^{3}w_{3}^{n−3} and w_{3}^{n−1} form an additive basis for H^{3n−3} (G_{3,n}; ℤ_{2}). Using the Gröbner basis G, we have

Sq^{1}(w_{1}w_{2}w_{3}^{n−2} ) = nw_{1}^{2}w_{2}w_{3}^{n−2} + w_{1}w_{3}^{n−1} = g_{1,n−2} + w_{2}^{2}w_{3}^{n−2} = w_{2}^{2}w_{3}^{n−2} ,
Sq^{1}(w_{2}^{3}w_{3}^{n−3} ) = nw_{1}w_{2}^{3}w_{3}^{n−3} + 3w_{2}^{2}w_{3}^{n−2} = w_{1}w_{2}^{3}w_{3}^{n−3} + w_{2}^{2}w_{3}^{n−2} = g_{3,n−3} = 0,
Sq^{1}(w_{3}^{n−1} ) = (n − 1)w_{1}w_{3}^{n−1} = 0,

and the lemma is proved. □

In the proof of the following lemma, we shall make use of the fact that for any cohomology class u and any nonnegative integers m and k,

Sq^{m}(u^{2^{k}} ) = \{ \begin{matrix}(Sq^{\frac{m}{2^{k}}} u)^{2^{k}} , & 2^{k} | m \\ 0, & 2^{k} ∤ m \end{matrix}

The case k = 1 is obtained from the Cartan formula and the rest is easily proved by induction on k.

Lemma 3.5. For the class w_{1}w_{2}^{4}w_{3}^{n−5} ∈ H^{3n−6} (G_{3,n}; ℤ_{2}), we have

(a) Sq^{2}Sq^{1}(w_{1}w_{2}^{4}w_{3}^{n−5} ) = w_{3}^{n−1} ,

(b) Sq^{2}(w_{1}w_{2}^{4}w_{3}^{n−5} ) = 0,

(c) (Sq^{4} + w_{2}^{2})(w_{1}w_{2}^{4}w_{3}^{n−5} ) = 0.

Proof. One has

Sq^{1}(w_{1}w_{2}^{4}w_{3}^{n−5} ) = nw_{1}^{2}w_{2}^{4}w_{3}^{n−5} + 4w_{1}w_{2}^{3}w_{3}^{n−4}
= w_{1}^{2}w_{2}^{4}w_{3}^{n−5} = g_{4,n−5} + w_{2}^{5}w_{3}^{n−5} = w_{2}^{5}w_{3}^{n−5} ;

and

Sq^{2}Sq^{1}(w_{1}w_{2}^{4}w_{3}^{n−5} ) = \binom{n}{2}w_{1}^{2}w_{2}^{5}w_{3}^{n−5} +5(n − 5)w_{1}w_{2}^{4}w_{3}^{n−4} + nw_{2}^{6}w_{3}^{n−5} +\binom{5}{2}w_{2}^{3}w_{3}^{n−3}
= w_{2}^{6}w_{3}^{n−5} = g_{6,n−5} + w_{3}^{n−1} = w_{3}^{n−1} .

This proves (a). Also,

Sq^{2}(w_{1}w_{2}^{4}w_{3}^{n−5} ) = \binom{n}{2}w_{1}^{3}w_{2}^{4}w_{3}^{n−5} + 4(n − 4)w_{1}^{2}w_{2}^{3}w_{3}^{n−4}
+ (n − 1)w_{1}w_{2}^{5}w_{3}^{n−5} + \binom{4}{2}w_{1}w_{2}^{2}w_{3}^{n−3}

and since n ≡ 1 (mod 8), this is obviously equal to zero.

Finally, for (c) we use the Cartan formula and we get

(Sq^{4} + w_{2}^{2})(w_{1}w_{2}^{4}w_{3}^{n−5} ) = w_{1}^{2}Sq^{3}(w_{2}^{4}w_{3}^{n−5} ) + w_{1}Sq^{4}(w_{2}^{4}w_{3}^{n−5} ) + w_{1}w_{2}^{6}w_{3}^{n−5} .

Now, since n − 5 is divisible by 4, w_{2}^{4}w_{3}^{n−5} = (w_{2}w3^{\frac{n−5}{4}} )^{4} and so Sq^{3}(w_{2}^{4}w_{3}^{n−5} ) = 0 and

Sq^{4}(w_{2}^{4}w_{3}^{n−5} ) = ( Sq^{1}( w_{2}w3^{\frac{n−5}{4}} ) )^{4} = ( (1 + \frac{n−5}{4} )w_{1}w_{2}w3^{\frac{n−5}{4}} + w_{3}^{\frac{n−5}{4} +1} )^{4} = w_{3}^{n−1}

where the latter equality holds because \frac{n−5}{4} is an odd integer (since n ≡ 1 (mod 8)). We conclude that

(Sq^{4} + w_{2}^{2})(w_{1}w_{2}^{4}w_{3}^{n−5} ) = w_{1}w_{3}^{n−1} + w_{1}w_{2}^{6}w_{3}^{n−5} = w_{1}g_{6,n−5} = 0,

and the proof of the lemma is completed. □

Lemma 3.6. For the classes w_{1}w_{2}^{2}w_{3}^{n−3} , w_{2}w_{3}^{n−2} ∈ H^{3n−4} (G_{3,n}; ℤ_{2}), we have

(a) Sq^{1}(w_{1}w_{2}^{2}w_{3}^{n−3} ) = w_{2}^{3}w_{3}^{n−3} + w_{3}^{n−1} , Sq^{1}(w_{2}w_{3}^{n−2} ) = w_{3}^{n−1} ;
(b) Sq^{2}(w_{1}w_{2}^{2}w_{3}^{n−3} + w_{2}w_{3}^{n−2} ) = 0.

Proof. (a) We have

Sq^{1}(w_{1}w_{2}^{2}w_{3}^{n−3} ) = nw_{1}^{2}w_{2}^{2}w_{3}^{n−3} + 2w_{1}w_{2}w_{3}^{n−2} = w_{1}^{2}w_{2}^{2}w_{3}^{n−3}
= g_{2,n−3} + w_{2}^{3}w_{3}^{n−3} + w_{3}^{n−1} = w_{2}^{3}w_{3}^{n−3} + w_{3}^{n−1} ,
Sq^{1}(w_{2}w_{3}^{n−2} ) = (n − 1)w_{1}w_{2}w_{3}^{n−2} + w_{3}^{n−1} = w_{3}^{n−1} .

(b) Similarly,

Sq^{2}(w_{1}w_{2}^{2}w_{3}^{n−3} +w_{2}w_{3}^{n−2} ) = \frac{n}{2} w_{1}^{3}w_{2}^{2}w_{3}^{n−3} +2(n−2)w_{1}^{2}w_{2}w_{3}^{n−2} +(n−1)w_{1}w_{2}^{3}w_{3}^{n−3}

+ \binom{2}{2}w_{1}w_{3}^{n−1} + \binom{n−1}{2}w_{1}^{2}w_{2}w_{3}^{n−2} + (n − 2)w_{1}w_{3}^{n−1} + (n − 1)w_{2}^{2}w_{3}^{n−2} = 0,

and we are done. □

Lemma 3.7. For the class w_{1}w_{3}^{n−2} ∈ H^{3n−5} (G_{3,n}; ℤ_{2}), we have

Sq^{2}(w_{1}w_{3}^{n−2} ) = w_{1}w_{2}w_{3}^{n−2} .

Proof. We simply calculate:

Sq^{2}(w_{1}w_{3}^{n−2} ) = \binom{n−1}{2}w_{1}^{3}w_{3}^{n−2} + (n − 2)w_{1}w_{2}w_{3}^{n−2} = w_{1}w_{2}w_{3}^{n−2} ,

and the lemma is proved. □

Proof of Theorem 1.1. It is well known that the Grassmann manifold G_{k,n} is orientable if and only if n + k is even, and therefore, G_{3,n} is orientable (the orientability of G_{3,n} can also be deduced from Lemma 3.1 (b)). We shall use the theorem of Hirsch [4] which states that a smooth orientable compact m-manifold M^{m} immerses into ℝ^{m+l} if and only if the classifying map f_{ν} : M^{m} → BSO of the stable normal bundle ν of M^{m} lifts up to BSO(l).

[figure: M^{m}; BSO; f_{ν}; BSO(l); BSO; p; M^{m}; BSO(l)]

The dimension of G_{3,n} is 3n, and hence, we need to lift f_{ν} : G_{3,n} → BSO up to BSO(3n − 6). The 3n-MPT for the fibration p: BSO(3n − 6) → BSO is given in Diagram 1 (K_{m} stands for the Eilenberg–MacLane space K(ℤ_{2}, m)).

The table of k-invariants is the following one.

k_{1}^{1} : (Sq^{2} + w_{2})w_{3n−5} = 0
k_{2}^{1} : (Sq^{2} + w_{2})Sq^{1}w_{3n−5} + Sq^{1}w_{3n−3} = 0
k_{3}^{1} : (Sq^{4} + w_{4})w_{3n−5} + Sq^{2}w_{3n−3} = 0
k_{1}^{2} : (Sq^{2} + w_{2})k_{1}^{1} + Sq^{1}k_{2}^{1} = 0

According to Lemma 3.1 (a), f_{ν}^{∗}(w_{3n−5} ) = w_{3n−5} (ν) = 0 and f_{ν}^{∗}(w_{3n−3} ) = w_{3n−3} (ν) = 0, so there is a lifting g_{1} : G_{3,n} → E_{1} of f_{ν}.

[figure: G_{3,n}; BSO; f_{ν}; BSO; K_{3n−5} × K_{3n−3}; w_{3n−5} ×w_{3n−3}; E_{1}; BSO; q_{1}; E_{1}; K_{3n−4} × K_{3n−3} × K_{3n−2}; k_{1}^{1} ×k_{2}^{1} ×k_{3}^{1}; E_{2}; E_{1}; q_{2}; E_{2}; K_{3n−3}; k_{1}^{2}; BSO(3n − 6); E_{2}; q_{3}; G_{3,n}; E_{1}; g; G_{3,n}; E_{2}; h] Diagram 1.

Let us remark here that for every lifting g: G_{3,n} → E_{1} of f_{ν}, one has

(3.3)  Sq^{2}(g^{∗}(k_{1}^{1})) = Sq^{1}(g^{∗}(k_{2}^{1})).

This is obtained by applying g^{∗} to the relation (Sq^{2}+w_{2})k_{1}^{1} = Sq^{1}k_{2}^{1} in H^{∗}(E_{1}; ℤ_{2}) (which produces the k-invariant k_{1}^{2}) and using Lemma 3.1 (b).

We have a lifting g_{1} : G_{3,n} → E_{1} and in order to make the next step (to lift f_{ν} up to E_{2}), we need to modify g_{1} (if necessary) to a lifting g such that g^{∗}(k_{1}^{1}) = g^{∗}(k_{2}^{1}) = g^{∗}(k_{3}^{1}) = 0. By choosing a map α×β: G_{3,n} → K_{3n−6} ×K_{3n−4} = Ω(K_{3n−5} ×K_{3n−3} ) (i.e., classes α ∈ H^{3n−6} (G_{3,n}; ℤ_{2}) and β ∈ H^{3n−4} (G_{3,n}; ℤ_{2})), we get another lifting g_{2} : G_{3,n} → E_{1} (induced by g_{1}, α and β) as the composition:

G_{3,n} ⟶^{△} G_{3,n} × G_{3,n} −−−−−−→^{(α×β)×g_{1}} K_{3n−6} × K_{3n−4} × E_{1} ⟶^{μ} E_{1}

where △ is the diagonal mapping and μ: Ω(K_{3n−5} ×K_{3n−3} )×E_{1} → E_{1} is the action of the fibre in the principal fibration q_{1} : E_{1} → BSO. By looking at the relations that produce the k-invariants k_{1}^{1}, k_{2}^{1} and k_{3}^{1} and using Lemma 3.1, we conclude that the following equalities hold (see [3, p. 95]):

g_{2}^{∗}(k_{1}^{1}) = g_{1}^{∗}(k_{1}^{1}) + (Sq^{2} + w_{2}(ν))(α) = g_{1}^{∗}(k_{1}^{1}) + Sq^{2}α;

g_{2}^{∗}(k_{2}^{1}) = g_{1}^{∗}(k_{2}^{1}) + (Sq^{2} + w_{2}(ν))Sq^{1}α + Sq^{1}β = g_{1}^{∗}(k_{2}^{1}) + Sq^{2}Sq^{1}α + Sq^{1}β;
g_{2}^{∗}(k_{3}^{1}) = g_{1}^{∗}(k_{3}^{1})+(Sq^{4}+w_{4}(ν))(α)+Sq^{2}β = g_{1}^{∗}(k_{3}^{1})+(Sq^{4}+w_{2}^{2})(α)+Sq^{2}β.

First, we need to prove that g_{1}^{∗}(k_{1}^{1}) is in the image of Sq^{2} : H^{3n−6} (G_{3,n}; ℤ_{2}) → H^{3n−4} (G_{3,n}; ℤ_{2}). Let us assume, to the contrary, that g_{1}^{∗}(k_{1}^{1}) is not in this image. The classes w_{1}^{2}w_{3}^{n−2} , w_{1}w_{2}^{2}w_{3}^{n−3} , w_{2}^{4}w_{3}^{n−4} and w_{2}w_{3}^{n−2} form a vector space basis for H^{3n−4} (G_{3,n}; ℤ_{2}) and from Lemma 3.2 we conclude that the sum of all basis elements and the sum of any two basis elements are in the image of Sq^{2}. This means that g_{1}^{∗}(k_{1}^{1}) is either a basis element or a sum of three distinct basis elements. Now, by looking at Lemma 3.3, we see that Sq^{2}(g_{1}^{∗}(k_{1}^{1})) ∈ {w_{1}w_{3}^{n−1} , w_{1}w_{3}^{n−1} + w_{2}^{2}w_{3}^{n−2} } and from formula (3.3) we have that Sq^{2}(g_{1}^{∗}(k_{1}^{1})) = Sq^{1}(g_{1}^{∗}(k_{2}^{1})). But according to Lemma 3.4 and the fact that the set {w_{1}w_{3}^{n−1} , w_{2}^{2}w_{3}^{n−2} } is a vector space basis for H^{3n−2} (G_{3,n}; ℤ_{2}), Sq^{1}(g_{1}^{∗}(k_{2}^{1})) cannot belong to {w_{1}w_{3}^{n−1} , w_{1}w_{3}^{n−1} + w_{2}^{2}w_{3}^{n−2} }. This contradiction proves that we can find a class α ∈ H^{3n−6} (G_{3,n}; ℤ_{2}) such that Sq^{2}α = g_{1}^{∗}(k_{1}^{1}).

Since {w_{1}w_{3}^{n−1} , w_{2}^{2}w_{3}^{n−2} } is a basis for H^{3n−2} (G_{3,n}; ℤ_{2}), by Lemma 3.3, there is a class β ∈ H^{3n−4} (G_{3,n}; ℤ_{2}) such that Sq^{2}β = g_{1}^{∗}(k_{3}^{1}) + (Sq^{4} + w_{2}^{2})(α), and so we have a lifting g_{2} : G_{3,n} → E_{1} (induced by g_{1} and these classes α and β) such that g_{2}^{∗}(k_{1}^{1}) = g_{2}^{∗}(k_{3}^{1}) = 0.

There is one more obstruction for lifting f_{ν} up to E_{2}: g_{2}^{∗}(k_{2}^{1}) ∈ H^{3n−3} (G_{3,n}; ℤ_{2}). Since g_{2}^{∗}(k_{1}^{1}) = 0, by equality (3.3) we have that Sq^{1}(g_{2}^{∗}(k_{2}^{1})) = 0 and according to Lemma 3.4, g_{2}^{∗}(k_{2}^{1}) must be in the subgroup of H^{3n−3} (G_{3,n}; ℤ_{2}) generated by w_{2}^{3}w_{3}^{n−3} and w_{3}^{n−1} . Observe the classes α′ := w_{1}w_{2}^{4}w_{3}^{n−5} ∈ H^{3n−6} (G_{3,n}; ℤ_{2}) and β′ := w_{1}w_{2}^{2}w_{3}^{n−3} + w_{2}w_{3}^{n−2} ∈ H^{3n−4} (G_{3,n}; ℤ_{2}). By Lemma 3.5 (a), Sq^{2}Sq^{1}α′ = w_{3}^{n−1} and according to Lemma 3.6 (a), Sq^{1}β′ = w_{2}^{3}w_{3}^{n−3} . This means that we can choose the coefficients a, b ∈ {0, 1} such that Sq^{2}Sq^{1}(aα′) + Sq^{1}(bβ′) = g_{2}^{∗}(k_{2}^{1}). Finally, from Lemma 3.5, parts (b) and (c), and Lemma 3.6(b), we conclude that for the lifting g: G_{3,n} → E_{1} induced by g_{2} and the classes aα′ and bβ′, all obstructions vanish, i.e., g^{∗}(k_{1}^{1}) = g^{∗}(k_{2}^{1}) = g^{∗}(k_{3}^{1}) = 0.

Therefore, the lifting g lifts up to E_{2}, i.e., there is a map h: G_{3,n} → E_{2} such that q_{1} ∘ q_{2} ∘ h = q_{1} ∘ g = f_{ν}.

For the final step, we observe that the set {w_{1}w_{2}w_{3}^{n−2} , w_{2}^{3}w_{3}^{n−3} , w_{3}^{n−1} } is a vector space basis for H^{3n−3} (G_{3,n}; ℤ_{2}). By looking at the relation that produces the k-invariant k_{1}^{2} and according to Lemma 3.6(a), Lemma 3.7 and Lemma 3.1(b), one sees that the indeterminacy of k_{1}^{2} is all of H^{3n−3} (G_{3,n}; ℤ_{2}). Hence, the lifting h: G_{3,n} → E_{2} can be chosen such that h^{∗}(k_{1}^{2}) = 0. This completes the proof of the theorem. □

Acknowledgments. The first author was partially supported by the Ministry of Science and Environmental Protection of the Republic of Serbia, Project #174032. The second author was partially supported by the Ministry of Science and Environmental Protection of the Republic of Serbia, Project #174034.

References

1. R. Cohen, The immersion conjecture for differentiable manifolds, Ann. Math. (2) 122 (1985) 237–328.

2. S. Dutta, S. S. Khare, On second Stiefel–Whitney class of Grassmann manifolds and cuplength, J. Indian Math. Soc. 69 (2002) 237–251.

3. S. Gitler, M. Mahowald, The geometric dimension of real stable vector bundles, Bol. Soc. Mat. Mex. 11 (1966) 85–107.

4. M. W. Hirsch, Immersions of manifolds, Trans. Am. Math. Soc. 93 (1959), 242–276.

5. V. Oproiu, Some non-embedding theorems for Grassmann manifolds G_{2,n} and G_{3,n} , Proc. Edinburgh Math. Soc. 20 (1976-77), 177–185.

6. Z. Z. Petrović, B. I. Prvulović, On Gröbner bases and immersions of Grassmann manifolds G_{2,n} , Homology Homotopy Appl. 13(2) (2011), 113–128.

7. , Note on immersion dimension of real Grassmannians, Topology Appl. 175 (2014), 38–42.

8. , Gröbner bases and some immersion theorems for Grassmann manifolds G_{3,n} , arXiv:1306.4814.

9. Z. Z. Petrović, B. I. Prvulović, M. Radovanović, Multiplication in the cohomology of Grassmannians via Gröbner bases, J. Algebra 438 (2015), 60–84.

10. R. E. Stong, Cup products in Grassmannians, Topology Appl. 13 (1982), 103–113.

University of Belgrade (Received 18 01 2016) Faculty of Mathematics Belgrade Serbia

zoranp@matf.bg.ac.rs
bane@matf.bg.ac.rs