GROEBNER BASES AND THE COHOMOLOGY OF GRASSMANN MANIFOLDS WITH APPLICATION TO IMMERSION

KENNETH G. MONKS

Abstract. Let G_{k,n} be the Grassmann manifold of k-planes in R^{n+k}. Borel showed that H^{∗} (G_{k,n}; Z_{2}) = Z_{2} [w_{1}, …, w_{k}] /I_{k,n} where I_{k,n} is the ideal generated by the dual Stiefel-Whitney classes \overline{w}_{n+1}, …, \overline{w}_{n+k}. We compute Groebner bases for the ideals I_{2,2^{i}−3} and I_{2,2^{i}−4} and use these results along with the theory of modified Postnikov towers to prove immersion results, namely that G_{2,2^{i}−3} immerses in R^{2^{i+2}−15}. As a benefit of the Groebner basis theory we also obtain a simple description of H^{∗} (G_{2,2^{i}−3}; Z_{2}) and H^{∗} (G_{2,2^{i}−4}; Z_{2}) and use these results to give a simple proof of some non-immersion results of Oproui.

1. Introduction

Let G_{k,n} denote the Grassmann manifold of unoriented k-planes in ℝ^{n+k}. Define the immersion dimension of G_{k,n} to be the smallest j such that G_{k,n} immerses in ℝ^{j}. There have been many results in the literature which obtain upper and lower bounds on the immersion dimension of G_{k,n}. Lower bounds are obtained for certain families of G_{k,n} in [6], [9], [12], [13] through the use of Stiefel-Whitney classes. Upper bounds have been obtained by Lam [7] who showed G_{k,n} has immersion dimension less than or equal to \binom{n+k}{2}. For many values of n, k Lam’s result improves on the standard upper bound of 2nk − 1 given by Whitney’s theorem [14] and the stronger upper bound of 2nk − α (nk) (where α (nk) is the number of ones in the binary expansion of nk) given by Cohen’s theorem [3]. However, except for the results of Hiller-Stong [6] who showed the immersion dimension G_{k,k} and G_{k,k+1} is equal to Lam’s upper bound, the exact immersion dimension of G_{k,n} is largely not known. There are also many results on the immersion and non-immersion of projective spaces which is the case G_{1,n} (or since G_{k,n} is diffeomorphic to G_{n,k}, we could also say G_{k,1}). As we are not interested in the projective space case in this article, we shall henceforth assume that 1 < k ≤ n when discussing G_{k,n}.

Many common techniques for computing bounds on the immersion dimension of G_{k,n} require a good working understanding of the structure of the cohomology ring H^{∗} (G_{k,n}; ℤ_{2}) . Borel [1] gave the following description of this ring. It is well known that

Date: March 31, 2001.

2000 Mathematics Subject Classification. Primary 57R42, 13P10, 57N65; Secondary 57R20, 55S45.

Supported in part by the C. C. Hsiung Visiting Associate Professorship at Lehigh University.

H^{∗} (G_{k,∞}; ℤ_{2}) = H^{∗} (BO(k); ℤ_{2}) = ℤ_{2} [w_{1}, …, w_{k}]

where w_{i} is the i^{th} Stiefel-Whitney class of the canonical k-bundle γ over BO (k) . Let \overline{w} = 1 + \overline{w}_{1} + \overline{w}_{2} + ⋯ and w = 1 + w_{1} + w_{2} + ⋯ satisfy the relation w\overline{w} = 1. Borel showed that

H^{∗} (G_{k,n}; ℤ_{2}) = ℤ_{2} [w_{1}, …, w_{k}] /I_{k,n}

where I_{k,n} = ⟨w̄_{1+n}, …, w̄_{k+n}⟩ . Chern [2] also gave a complete description of this ring involving Schubert symbols.

However both descriptions are often quite unwieldy to work with in practice. This lack of a simple workable description has been one of the major obstacles to computing immersion bounds (e.g. when computing the Stiefel-Whitney classes of an arbitrary Grassmann). In this paper we apply the methods of Groebner bases (c.f. [4]) to Borel’s description of the cohomology to give a very tractable and simple description of the cohomology of certain Grassmanns and then use this to obtain new immersion results. We also illustrate how the same techniques can be used to reprove in a straightforward manner some nonimmersion results obtained by Oproui in [12].

2. Main Results

Throughout the paper we fix a monomial ordering on the monomials in ℤ_{2} [w_{1}, w_{2}] by defining w_{2}^{b}w_{1}^{a} ≺ w_{2}^{d}w_{1}^{c} if either b < d or else b = d and a < c. We prove the following. Theorem 2.1. With respect to the ordering ≺, and for i ≥ 3,

(1) \{\overline{w}_{2^{i}−2}, \overline{w}_{2^{i}−1}\} is the reduced Groebner basis for I_{2,2^{i}−3}.

(2) \{\overline{w}_{2^{i}−3}, \overline{w}_{2^{i}−2}, \overline{w}_{2^{i}−1}\} is the reduced Groebner basis for I_{2,2^{i}−4}.

From this we obtain as a corollary the following simple descriptions of the cohomology in these cases. Corollary 2.2. For i ≥ 3,

(1) A vector space basis for H^{∗} (G_{2,2^{i}−3}; ℤ_{2}) is the set of all monomials w_{2}^{b}w_{1}^{a} such that a < 2^{i} − 1 and b < 2^{i−1} − 1. The product structure is completely determined by the relations w_{1}^{2^{i}−1} = 0 and w_{2}^{2^{i−1}−1} = ∑_{j=0}^{i−2} w_{2}^{2^{j}−1}w_{1}^{2^{i}−2^{j+1}}.

(2) A vector space basis for H^{∗} (G_{2,2^{i}−4}; ℤ_{2}) is the set of all monomials w_{2}^{b}w_{1}^{a}

such that a < 2^{i} − 1 and b < 2^{i−1} − 2 union with {w_{2}^{2^{i−1}−2}} . The prod-

uct structure is completely determined by the relations w_{1}^{2^{i}−1} = 0, w_{2}^{2^{i−1}−1} = ∑_{j=0}^{i−2} w_{2}^{2^{j}−1}w_{1}^{2^{i}−2^{j+1}}, and w_{2}^{2^{i−1}−2}w_{1} = ∑_{j=1}^{i−2} w_{2}^{2^{j}−2}w_{1}^{2^{i}−2^{j+1}+1}.

To illustrate these results, consider the case i = 3. By the corollary, H^{∗} (G_{2,5}; ℤ_{2}) has a basis consisting of all w_{2}^{b}w_{1}^{a} with b < 3 and a < 7 subject to the relations w_{1}^{7} = 0 and w_{2}^{3} = w_{2}w_{1}^{4} + w_{1}^{6}. Since these relations are obtained from the Groebner basis we can easily compute arbitrary products in this ring by the standard techniques described in [4]. For example, suppose we wanted to know if (w_{2})^{4} is zero in H^{∗} (G_{2,5}; ℤ_{2}) . We simply compute

(w_{2})^{4} = (w_{2}^{3}) w_{2}
= (w_{2}w_{1}^{4} + w_{1}^{6}) w_{2}
= w_{2}^{2}w_{1}^{4} + w_{2}w_{1}^{6}.

Since w_{2}^{2}w_{1}^{4} + w_{2}w_{1}^{6} is expressed in terms of our vector space basis, we see that (w_{2})^{4} is not zero, and we have also determined its unique representation in our basis.

As a further application of these results, we obtain a new immersion result. Cohen’s result shows G_{2,2^{i}−3} immerses in ℝ^{2^{i+2}−13−i}. A scan of the literature indicates this is the smallest previously known upper bound. Using the description of the cohomology given above, and the method of modified Postnikov towers as described by Gitler and Mahowald [5] and extended by Nussbaum [11], we improve on this bound with the following new immersion result. Theorem 2.3. For i ≥ 3, G_{2,2^{i}−3} immerses in ℝ^{2^{i+2}−15}.

Comparing with Whitney’s result, we see that our result improves on his upper bound by two dimensions for all i ≥ 3. Comparing with Cohen’s result, we see that our result improves on his upper bound by one dimension when i = 3. Theorem 2.3 shows G_{2,5} immerses in ℝ^{17}, whereas Cohen’s result shows it immerses in ℝ^{18}.

As another application of these results we can easily obtain lower bounds on immersion using Stiefel-Whitney classes. Theorem 2.4 (Oproui). For i ≥ 3,

(1) G_{2,2^{i}−3} does not immerse in ℝ^{2^{i+1}−3}.

(2) G_{2,2^{i}−4} does not immerse in ℝ^{2^{i+1}−3}.

These results were originally proven by Oproui [12], and later improved on for i > 3 by Tang [15] using completely different methods. But our use of Groebner bases allows us to give a proof which is quite straightforward.

3. Background

3.1. Background on Groebner bases. All of the information in this section can be found in [4].

A monomial ordering on ℤ_{2}[x_{1}, …, x_{k}] is a total well-ordering ≺ on the set of monomials such that x ≺ y implies xz ≺ yz for any monomials x, y, z. For example, lexicographic ordering with respect to some ordering of the variables is a monomial ordering. Let ≺ be a monomial ordering on ℤ_{2}[x_{1}, …, x_{k}] and f ∈ ℤ_{2}[x_{1}, …, x_{k}]. Then LT (f) denotes the leading term of f with respect to ≺, i.e. the largest monomial summand of f with respect to ≺ . Let F = (f_{1}, …, f_{s}) be an ordered s-tuple of polynomials in ℤ_{2}[x_{1}, …, x_{k}]. There is a division algorithm which allows us to write every f ∈ ℤ_{2}[x_{1}, …, x_{k}] as

f = a_{1}f_{1} + a_{2}f_{2} + ⋯ a_{s}f_{s} + r

where a_{i}, r ∈ ℤ_{2}[x_{1}, …, x_{k}] and r is a ℤ_{2}-linear combination of monomials, none of which is divisible by any of LT (f_{1}) , …, LT (f_{s}) . This remainder r is sometimes denoted \overline{f}^{F} . It is not necessarily unique in the sense that changing the order of the functions in F can produce a different remainder. Even worse is the fact that f ∈ I = ⟨f_{1}, …, f_{s}⟩ does not imply the remainder, r, on dividing f by F is necessarily 0.

Let I ⊆ ℤ_{2}[x_{1}, …, x_{k}] be a non-zero ideal. Let LT (I) denote the set of all monomials which are leading terms of elements of I. Fix a monomial order. A finite subset G = {g_{1}, …, g_{t}} of an ideal I is said to be a Groebner basis for I if ⟨LT (g_{1}) , LT (g_{2}) , …, LT (g_{t})⟩ = ⟨LT (I)⟩ . A Groebner basis is reduced if for every g_{j} ∈ G, no summand of g_{j} is divisible by any LT (g_{i}) with i ≠ j.

Let G = {g_{1}, …, g_{t}} be a Groebner basis for I and f ∈ ℤ_{2}[x_{1}, …, x_{k}]. Then there is a unique r ∈ ℤ_{2}[x_{1}, …, x_{k}] such that no term of r is divisible by one of LT (g_{1}) , LT (g_{2}) , …, LT (g_{t}) and f = g + r for some g ∈ I. In addition, f ∈ I if and only if r = 0. Then ℤ_{2}[x_{1}, …, x_{k}]/I is isomorphic as a ℤ_{2} vector space to

Span {x : x is a monomial and x ∉ ⟨LT (I)⟩} ,

hence the set of monomials which are not divisible by any of LT (g_{1}) , LT (g_{2}) , …, LT (g_{t}) is a basis for the quotient ring (i.e. it is a set of canonical equivalence class representatives).

Let f, g ∈ ℤ_{2}[x_{1}, …, x_{k}] be nonzero polynomials. The S-polynomial of f and g is

S (f, g) = \frac{L}{LT (f)} · f + \frac{L}{LT (g)} · g

where L = LCM (LT (f) , LT (g)) . Buchberger’s algorithm states that a basis G = {g_{1}, …, g_{t}} for I is a Groebner basis if and only if for all pairs i ≠ j the remainder on division of S (g_{i}, g_{j}) by G is zero.

3.2. Background on Stiefel-Whitney classes and non-immersion. Most of the following background information can be found in [10] and [7].

Let M^{m} denote an m-dimensional paracompact Hausdorff smooth connected manifold. Define

imm (M^{m}) = min {j : M^{m} immerses in ℝ^{j}} .

Let w_{j} (M^{m}) denote the j^{th} Stiefel-Whitney class of the manifold, i.e. the j^{th} Stiefel-Whitney class of the tangent bundle τ. Let \overline{w}_{j} (M^{m}) denote the j^{th} dual Stiefel-Whitney class of M^{m}, i.e. the j^{th} Stiefel-Whitney class of the stable normal bundle ν. If we let w (M^{m}) = 1 + w_{1} (M^{m}) + w_{2} (M^{m}) + ⋯ ∈ H^{∗} (M^{m}; ℤ_{2}) denote the total Stiefel-Whitney class and \overline{w} (M^{m}) = 1 + \overline{w}_{1} (M^{m}) + \overline{w}_{2} (M^{m}) + ⋯ ∈ H^{∗} (M^{m}; ℤ_{2}) denote the total dual Stiefel-Whitney of M^{m} then w (M^{m}) \overline{w} (M^{m}) = 1. If w̄_{j} (M^{m}) ≠ 0 then m + j ≤ imm (M^{m}) . Thus to apply this result to Grassmann manifolds we need information regarding ν.

Let γ denote the canonical vector bundle of dimension k over G_{k,n} consisting of those points (x, v) such that x ∈ G_{k,n} and v ∈ x. Let γ^{⊥} denote the orthogonal bundle of dimension n over G_{k,n} consisting of those points (x, v) such that x ∈ G_{k,n} and v ⊥ x. Then γ ⊕γ^{⊥} = (n + k) ε where ε denotes the trivial line bundle over G_{k,n}. Let τ denote the tangent bundle of G_{k,n.} It is well known (c.f. [6]) that τ = γ ⊗ γ^{⊥} and thus

τ ⊕ γ ⊗ γ = γ ⊗ γ^{⊥} ⊕ γ ⊗ γ
= γ ⊗ (γ^{⊥} ⊕ γ)
= γ ⊗ (n + k) ε
= (n + k) γ.

Computing the Stiefel-Whitney classes of both sides of the equation and solving for 1/w (G_{k,n}) shows that the total dual Stiefel Whitney class of G_{k,n} is given by

\overline{w} (G_{k,n}) = \frac{w (γ ⊗ γ)}{w (γ)^{n+k}}.

Now w (γ) = 1 + w_{1} + w_{2} + ⋯ + w_{k}. Using the formula for the Stiefel-Whitney class of a tensor product shows that for k = 2, w (γ ⊗ γ) = 1 + w_{1}^{2}. Thus

\overline{w} (G_{2,n}) = \frac{1 + w_{1}^{2}}{(1 + w_{1} + w_{2})^{n+2}} .

Thus we can obtain lower bounds on the immersion dimension of G_{2,n} by computing the highest grading in which \frac{1+w_{1}^{2}}{(1+w_{1}+w_{2})^{n+2}} is nonzero in the cohomology of G_{2,n}.

3.3. Background on Modified Postnikov Towers and immersion. Most of the following information can be found in [5].

The method of Gitler and Mahowald begins with an important theorem of Hirsch

Theorem 3.1 (Hirsch 1959). The following are equivalent.

(1) M^{m} immerses in ℝ^{m+p}.

(2) M^{m} has a normal bundle ν which is a p-plane bundle.

(3) There is a lifting ν_{p}

[figure: M^{m}; BO; ν; q; BO(p); ν_{p}]

of the classifying map ν of the stable normal bundle of M^{m}. The obstructions to such a lifting are classes in H^{∗} (M^{m}; π_{∗−1} (F)) where F is the

fibre of q. It is known that F = V_{p}, the Stiefel manifold of codimension p frames in ℝ^{∞}. This has the same mod 2 cohomology ring as the stunted projective space P_{p} thru grading 2p. Gitler and Mahowald’s method tries to construct such liftings by handling the obstructions one Adams filtration of π_{∗} (F) at a time. Nussbaum [11] showed that their method could be applied to the fibration BO (n) → BO when n is odd.

Thus by Hirsch’s theorem one can show imm (G_{k,n}) ≤ nk + j if we can lift the classifying map for the normal bundle to BO (j) . In particular we will lift the normal bundle map for G_{2,2^{i}−3} to BO (2^{i+1} − 9) to obtain our new results.

Finally, we will require Wu’s formula for the action of the Steenrod algebra on H^{∗} (BO; ℤ_{2}) = ℤ_{2}[w_{1}, w_{2}, …], namely

Sq^{k}w_{m} = ∑_{i=0}^{k} \binom{m + i − k − 1}{i}w_{k−i}w_{m+i}.

4. Proofs

4.1. Groebner bases.

4.1.1. Formulas for \overline{w}_{i}. The elements \overline{w}_{i} are defined in the Borel description above, by the relation w\overline{w} = 1. It is a simple matter using this relation to prove the following formulas for \overline{w}_{i} which can also be found in [10]. From now on we will restrict to the case k = 2. In ℤ_{2} [w_{1}, w_{2}] we have the closed formula

(1)  \overline{w}_{i} = ∑_{a+2b=i} \binom{a + b}{a}w_{2}^{b}w_{1}^{a}

where the binomial coefficients are considered mod 2. Alternatively, we have a recursive formula \overline{w}_{0} = 1, \overline{w}_{1} = w_{1,} and

(2)  \overline{w}_{i} = w_{1}\overline{w}_{i−1} + w_{2}\overline{w}_{i−2}  for i ≥ 1.

Using these results we can prove the following technical lemmas. However, we first need to review some facts about binomial coefficients mod 2.

For any positive integer a, let α_{i}(a) ∈ {0, 1} be the coefficient of 2^{i} in the binary expansion of a, i.e. a = ∑_{i=0}^{∞} α_{i}(a)2^{i}. We say that a and b are disjoint if α_{i}(a) + α_{i}(b) ≤ 1 for all i. By a famous result of Lucas [8], a and b are disjoint if and only if the binomial coefficient \binom{a+b}{b} is odd . We will also write 2^{i} ∈ a or say a contains 2^{i} as synonyms for α_{i}(a) = 1 since the meaning will be clear from the context.

Note that since 2^{i} − 1 = ∑_{j=0}^{i−1} 2^{j}, any integer of the form 2^{i} − 1 contains all 2^{j} with j < i. Additionally if s + t = 2^{i} − 1 then s and t must be disjoint. (To see this let 2^{m} be the smallest two power they have in common. Then by the algorithm for base two addition, 2^{m} will not be in their sum, 2^{i} − 1. But since 2^{m} ≤ s ≤ s + t = 2^{i} − 1, we see that 2^{m} must be in 2^{i} − 1 giving a contradiction.) Moreover, if s + t = 2^{i} − 1 then since s and t are disjoint, there can be no carries in the base two addition of s and t, so that for every j < i either 2^{j} ∈ s or 2^{j} ∈ t. Notice also that 2^{j} is in r if and only if 2^{j+1} is in 2r so that a and b are disjoint if and only if 2^{j}a and 2^{j}b are disjoint for any j. We will use such facts frequently in what follows. Lemma 4.1. In ℤ_{2} [w_{1}, w_{2}] for i ≥ 0 we have

\overline{w}_{2^{i}−1} = w_{1}^{2^{i}−1}.

Proof. By (1),

\overline{w}_{2^{i}−1} = ∑_{a+2b=2^{i}−1} \binom{a + b}{a}w_{1}^{a}w_{2}^{b}.

Suppose a + 2b = 2^{i} − 1. Then a and 2b are disjoint and if j < i then 2^{j} is contained in either a or 2b. Suppose b ≠ 0. Then let 2^{m} be the smallest two power in 2b. Since 2b is even, we have m > 0. Thus 2^{m−1} is not in 2b since m is the smallest. So 2^{m−1} is in a. But then 2^{m−1} is in b because 2^{m} is in 2b. So a and b are not disjoint. So \binom{a+b}{a} is zero mod 2. Thus the only summand on the right hand side of (1) which has nonzero coefficient occurs when b = 0, i.e. w_{1}^{2^{i}−1}. □ Lemma 4.2. In ℤ_{2} [w_{1}, w_{2}] for i ≥ 1 we have

\overline{w}_{2^{i}−2} = ∑_{j=0}^{i−1} w_{2}^{2^{j}−1}w_{1}^{2^{i}−2^{j+1}}.

Proof: By (1) and Lucas theorem it suffices to show that if a + 2b = 2^{i} − 2 then a and b are disjoint if and only if for some 0 ≤ j < i, b = 2^{j} − 1.

Suppose a + 2b = 2^{i} − 2 and for some 0 ≤ j < i, b = 2^{j} − 1. Then solving for a we get a = 2^{j+1} (2^{i−j−1} − 1) . So 2^{k} ∈ a implies k > j while 2^{k} ∈ b implies k < j, so a and b are disjoint.

Now suppose a + 2b = 2^{i} − 2 and a and b are disjoint. If b = 0 then b = 2^{0} − 1. So assume b ≠ 0. Let 2^{j−1} be the largest 2 power in b. Suppose there is h < j − 1 such that 2^{h} ∉ b and let m be the largest such h. Then 2^{m+1} ∈ b. Since 2b and 2^{i} − 2 are even so is a. Let A = a/2. Then A + b = 2^{i−1} − 1, so A and b are disjoint and for all l < i − 1, 2^{l} ∈ A or 2^{l} ∈ b. Thus 2^{m} ∈ b or 2^{m} ∈ A. But 2^{m} ∉ b by assumption so that 2^{m} ∈ A. But a = 2A so that 2^{m+1} ∈ a. So 2^{m+1} ∈ a and 2^{m+1} ∈ b, but this contradicts the fact that a and b are disjoint. Thus our supposition is false and so for all h ≤ j − 1, 2^{h} ∈ b. Thus b = 2^{j} − 1. □ Lemma 4.3. In ℤ_{2} [w_{1}, w_{2}] for i ≥ 2 we have

\overline{w}_{2^{i}−3} = ∑_{j=1}^{i−1} w_{2}^{2^{j}−2}w_{1}^{2^{i}−2^{j+1}+1}.

Proof: By (2) we have

\overline{w}_{2^{i}−1} = w_{1}\overline{w}_{2^{i}−2} + w_{2}\overline{w}_{2^{i}−3}.

Solving for \overline{w}_{2^{i}−3} and substituting for \overline{w}_{2^{i}−2} and \overline{w}_{2^{i}−1} via Lemma 4.2 and Lemma 4.1 gives us

\overline{w}_{2^{i}−3} = \frac{w_{1}\overline{w}_{2^{i}−2} + \overline{w}_{2^{i}−1}}{w_{2}}

= \frac{w_{1} ∑_{j=0}^{i−1} w_{2}^{2^{j}−1}w_{1}^{2^{i}−2^{j+1}} + w_{1}^{2^{i}−1}}{w_{2}}

= \frac{w_{1} (w_{1}^{2^{i}−2} + ∑_{j=1}^{i−1} w_{2}^{2^{j}−1}w_{1}^{2^{i}−2^{j+1}}) + w_{1}^{2^{i}−1}}{w_{2}}

= ∑_{j=1}^{i−1} w_{2}^{2^{j}−2}w_{1}^{2^{i}−2^{j+1}+1}.

□

4.1.2. Bases for I_{2,2^{i}−3} and I_{2,2^{i}−4}. Before proving Theorem 2.1, we prove a technical lemma. Lemma 4.4. Let x, y ∈ G ⊂ ℤ_{2}[x_{1}, …, x_{k}]. If x is a monomial and gcd (x, LT (y)) = 1 then \overline{S (x, y)}^{G} = 0.

Proof. If gcd (x, LT (y)) = 1 then S (x, y) = x LT (y) + xy. Since every summand of x LT (y) + xy is divisible by the monomial x, the remainder on division by G is zero. □

We are now ready to prove Theorem 2.1 which we restate here for convenience. Theorem 2.1. With respect to the ordering ≺, and for i ≥ 3,

(1) \{\overline{w}_{2^{i}−2}, \overline{w}_{2^{i}−1}\} is the reduced Groebner basis for I_{2,2^{i}−3}.

(2) \{\overline{w}_{2^{i}−3}, \overline{w}_{2^{i}−2}, \overline{w}_{2^{i}−1}\} is the reduced Groebner basis for I_{2,2^{i}−4}.

Proof: First we consider I_{2,2^{i}−3.} By Borel’s result G = \{\overline{w}_{2^{i}−2}, \overline{w}_{2^{i}−1}\} is a basis for I_{2,2^{i}−3}. To show it is Groebner we use Buchberger’s algorithm and compute \overline{S (\overline{w}_{2^{i}−2}, \overline{w}_{2^{i}−1})}^{G}. By Lemma 4.2 and Lemma 4.1 we see that LT (\overline{w}_{2^{i}−2}) = w_{2}^{2^{i−1}−1} and LT (\overline{w}_{2^{i}−1}) = w_{1}^{2^{i}−1}. Then gcd (w_{1}^{2^{i}−1}, w_{2}^{2^{i−1}−1}) = 1 and \overline{w}_{2^{i}−1} is a monomial, so

by Lemma 4.4, \overline{S (\overline{w}_{2^{i}−2}, \overline{w}_{2^{i}−1})}^{G} = 0. So by Buchberger’s algorithm G is Groebner.

To see that it is the reduced Groebner basis, it is a simple matter to check by Lemma 4.1 and Lemma 4.2 that no summand of \overline{w}_{2^{i}−2} is divisible by LT (\overline{w}_{2^{i}−1}) = w_{1}^{2^{i}−1}.

This completes the proof of the first part of the theorem. Note that this proof works for i = 2 as well.

Now let us consider I_{2,2^{i}−4}. By Borel’s result \{\overline{w}_{2^{i}−3}, \overline{w}_{2^{i}−2}\} is a basis for I_{2,2^{i}−4}. By Lemmas 4.2 and 4.3 above we have LT (\overline{w}_{2^{i}−3}) = w_{2}^{2^{i−1}−2}w_{1} and LT (\overline{w}_{2^{i}−2}) = w_{2}^{2^{i−1}−1}. Computing the S-polynomial of these two elements gives

S (\overline{w}_{2^{i}−3}, \overline{w}_{2^{i}−2}) = w_{2}\overline{w}_{2^{i}−3} + w_{1}\overline{w}_{2^{i}−2}
= \overline{w}_{2^{i}−1}
= w_{1}^{2^{i}−1}

by (2) and Lemma 4.1. Since neither leading term of \overline{w}_{2^{i}−3}, \overline{w}_{2^{i}−2} divides into w_{1}^{2^{i}−1} we must add this to our basis. Thus we let G = \{\overline{w}_{2^{i}−3}, \overline{w}_{2^{i}−2}, \overline{w}_{2^{i}−1}\} and we want to show this is Groebner. So we check the remaining S-polynomials.

Since gcd (w_{1}^{2^{i}−1}, w_{2}^{2^{i−1}−1}) = 1 we can apply Lemma 4.4 to conclude

\overline{S (\overline{w}_{2^{i}−2}, \overline{w}_{2^{i}−1})}^{G} = 0.

Next we need to consider

S (\overline{w}_{2^{i}−3}, \overline{w}_{2^{i}−1}) = w_{2}^{2^{i−1}−2}\overline{w}_{2^{i}−1} + w_{1}^{2^{i}−2}\overline{w}_{2^{i}−3}

= w_{2}^{2^{i−1}−2}w_{1}^{2^{i}−1} + w_{1}^{2^{i}−2} ∑_{j=1}^{i−1} w_{2}^{2^{j}−2}w_{1}^{2^{i}−2^{j+1}+1}

= ∑_{j=1}^{i−2} w_{2}^{2^{j}−2}w_{1}^{2^{i}−2^{j+1}+2^{i}−1}

= w_{1}^{2^{i}−1} ∑_{j=1}^{i−2} w_{2}^{2^{j}−2}w_{1}^{2^{i}−2^{j+1}}.

Thus every term of this S-polynomial is divisible by \overline{w}_{2^{i}−1} = w_{1}^{2^{i}−1}, so

\overline{S (\overline{w}_{2^{i}−3}, \overline{w}_{2^{i}−1})}^{G} = 0

. So G is a Groebner basis.

To see it is reduced, it is again a simple matter of checking that for i ≥ 3 that no term of any element of G is divisible by the leading term of any other element of G. □

Note that Corollary 2.2 follows immediately from Theorem 2.1 by solving the elements of the Groebner bases we found in each case for their leading terms (which is how they are used in practice).

4.2. Non-immersion results. In this section we illustrate how we can use the cohomology information given in Corollary 2.2 to easily reprove the non-immersion results of Oproui given in Theorem 2.4.

Theorem 2.4 (Oproui). For i ≥ 3,

(1) G_{2,2^{i}−3} does not immerse in ℝ^{2^{i+1}−3}.

(2) G_{2,2^{i}−4} does not immerse in ℝ^{2^{i+1}−3}.

Proof. As G_{2,2^{i}−3} is a submanifold of G_{2,2^{i}−4}, part 1 of the theorem follows immediately from part 2. However we prove both directly to illustrate the method of using Groebner bases in each case.

We first consider G_{2,2^{i}−3}. By the discussion in §3.2 we want to determine the highest grading in H^{∗} (G_{2,2^{i}−3}; ℤ_{2}) for which

\overline{w} (G_{2,2^{i}−3}) = \frac{1 + w_{1}^{2}}{(1 + w_{1} + w_{2})^{2^{i}−1}}

is nonzero. But H^{j} (G_{2,2^{i}−3}; ℤ_{2}) = 0 for j > 2^{i+1} − 6 so (1 + w_{1} + w_{2})^{2^{i+1}} = 1 in the cohomology ring. So we have

\frac{1 + w_{1}^{2}}{(1 + w_{1} + w_{2})^{2^{i}−1}} = \frac{1 + w_{1}^{2}}{(1 + w_{1} + w_{2})^{2^{i}−1}} (1 + w_{1} + w_{2})^{2^{i+1}}
= (1 + w_{1}^{2}) (1 + w_{1} + w_{2})^{2^{i}+1}
= (1 + w_{1}^{2}) (1 + w_{1}^{2^{i}} + w_{2}^{2^{i}}) (1 + w_{1} + w_{2}) .

Now the terms w_{1}^{2^{i}} and w_{2}^{2^{i}} are not in our vector space basis, so we convert them using our Groebner basis.

w_{1}^{2^{i}} = (w_{1}^{2^{i}−1}) w_{1} = 0 · w_{1} = 0,

and

w_{2}^{2^{i}} = (w_{2}^{2^{i−1}−1})^{2} w_{2}^{2}

= (∑_{j=0}^{i−2} w_{2}^{2^{j}−1}w_{1}^{2^{i}−2^{j+1}})^{2} w_{2}^{2}

= ∑_{j=0}^{i−2} w_{2}^{2^{j+1}}w_{1}^{2^{i+1}−2^{j+2}}

= (w_{1}^{2^{i}−1}) ∑_{j=0}^{i−2} w_{2}^{2^{j+1}}w_{1}^{2^{i}−2^{j+2}+1}

= 0 · ∑_{j=0}^{i−2} w_{2}^{2^{j+1}}w_{1}^{2^{i}−2^{j+2}+1}

= 0.

Thus we see that

\overline{w} (G_{2,2^{i}−3}) = (1 + w_{1}^{2}) (1 + w_{1} + w_{2})
= 1 + w_{1} + w_{2} + w_{1}^{2} + w_{1}^{3} + w_{2}w_{1}^{2}.

Since i > 2, these are all elements of the vector space basis, so that the highest grading nonzero term is w_{2}w_{1}^{2} in grading 4. Thus since the dimension of G_{2,2^{i}−3} is 2^{i+1} − 6, we see that imm (G_{2,2^{i}−3}) ≥ 2^{i+1} − 2.

Now we consider G_{2,2^{i}−4}. We want to determine the highest grading in H^{∗} (G_{2,2^{i}−4}; ℤ_{2}) for which

\overline{w} (G_{2,2^{i}−4}) = \frac{1 + w_{1}^{2}}{(1 + w_{1} + w_{2})^{2^{i}−2}}

is nonzero. But H^{j} (G_{2,2^{i}−4}; ℤ_{2}) = 0 for j > 2^{i+1} − 8 so again (1 + w_{1} + w_{2})^{2^{i+1}} = 1 in the cohomology ring. So we have

\frac{1 + w_{1}^{2}}{(1 + w_{1} + w_{2})^{2^{i}−2}} = \frac{1 + w_{1}^{2}}{(1 + w_{1} + w_{2})^{2^{i}−2}} (1 + w_{1} + w_{2})^{2^{i+1}}
= (1 + w_{1}^{2}) (1 + w_{1} + w_{2})^{2^{i}+2}
= (1 + w_{1}^{2}) (1 + w_{1}^{2^{i}} + w_{2}^{2^{i}}) (1 + w_{1}^{2} + w_{2}^{2}) .

Now the terms w_{1}^{2^{i}} and w_{2}^{2^{i}} are not in our vector space basis, so we convert them using our Groebner basis. The calculation goes through exactly as before and we find that w_{1}^{2^{i}} = w_{2}^{2^{i}} = 0. Thus we see that

\overline{w} (G_{2,2^{i}−4}) = (1 + w_{1}^{2}) (1 + w_{1}^{2} + w_{2}^{2})
= 1 + w_{2}^{2} + w_{1}^{4} + w_{2}^{2}w_{1}^{2}.

For i > 3, these are all elements of the basis, so that the highest grading term is w_{2}^{2}w_{1}^{2} in grading 6. Thus since the dimension of G_{2,2^{i}−4} is 2^{i+1} − 8, we see that imm (G_{2,2^{i}−4}) ≥ 2^{i+1} − 2.

For i = 3 the term w_{2}^{2}w_{1}^{2} is no longer in the basis. Dividing w_{2}^{2}w_{1}^{2} by our Groebner basis gives

w_{2}^{2}w_{1}^{2} = (w_{2}^{2}w_{1}) w_{1} = (w_{1}^{5}) w_{1} = w_{1}^{6}

which is in the basis, so that the result holds for i = 3 as well. □

4.3. Immersion Results. In this section we use the technique of modified Postnikov towers described in Gitler-Mahowald [5] as extended by Nussbaum [11] to prove Theorem 2.3. Before proving the theorem, we require the following technical lemmas which are consequences of Wu’s formula.

Lemma 4.5. Let m ≥ 0. Then in H^{∗} (BO (2) ; ℤ_{2}) = ℤ_{2} [w_{1}, w_{2}] we have

(1) Sq^{1}w_{1}^{m} = \{ \begin{matrix}0 & if m is even \\ w_{1}^{m+1} & if m is odd \end{matrix} .

(2) Sq^{1}w_{2}^{m} = \{ \begin{matrix}0 & if m is even \\ w_{2}^{m}w_{1} & if m is odd \end{matrix} .

(3) Sq^{2}w_{1}^{m} = \{ \begin{matrix}0 & if m ≡ 0, 1 mod 4 \\ w_{1}^{m+2} & if m ≡ 2, 3 mod 4 \end{matrix} .

(4) Sq^{2}w_{2}^{m} = \{ \begin{matrix}0 & if m ≡ 0 mod 4 \\ w_{2}^{m+1} & if m ≡ 1 mod 4 \\ w_{2}^{m}w_{1}^{2} & if m ≡ 2 mod 4 \\ w_{2}^{m+1} + w_{2}^{m}w_{1}^{2} & if m ≡ 3 mod 4 \end{matrix} .

Each formula can be proven by induction on m using the Wu formula and the Cartan formula. The details are elementary and straightforward, so we omit them here.

Lemma 4.6. For i ≥ 3, Sq^{2}Sq^{1} (w_{2}^{2^{i−1}−2}w_{1}^{2^{i}−5}) = w_{2}^{2^{i−1}−2}w_{1}^{2^{i}−2}.

Proof: Applying the previous lemma and the Cartan formula,

Sq^{1} (w_{2}^{2^{i−1}−2}w_{1}^{2^{i}−5}) = Sq^{1}w_{2}^{2^{i−1}−2} · w_{1}^{2^{i}−5} + w_{2}^{2^{i−1}−2} · Sq^{1}w_{1}^{2^{i}−5}
= 0 · w_{1}^{2^{i}−5} + w_{2}^{2^{i−1}−2} · w_{1}^{2^{i}−4}
= w_{2}^{2^{i−1}−2}w_{1}^{2^{i}−4}.

Thus

Sq^{2}Sq^{1} (w_{2}^{2^{i−1}−2}w_{1}^{2^{i}−5}) = Sq^{2} (w_{2}^{2^{i−1}−2}w_{1}^{2^{i}−4})
= Sq^{2}w_{2}^{2^{i−1}−2} · w_{1}^{2^{i}−4} + Sq^{1}w_{2}^{2^{i−1}−2} · Sq^{1}w_{1}^{2^{i}−4}
+ w_{2}^{2^{i−1}−2} · Sq^{2}w_{1}^{2^{i}−4}
= w_{2}^{2^{i−1}−2}w_{1}^{2} · w_{1}^{2^{i}−4} + 0 · 0 + w_{2}^{2^{i−1}−2} · 0
= w_{2}^{2^{i−1}−2}w_{1}^{2^{i}−2}.
□

Theorem 2.3. For i ≥ 3, G_{2,2^{i}−3} immerses in ℝ^{2^{i+2}−15}.

Proof: To simplify the notation we let m = 2^{i−2} − 2. As discussed in the background section, it suffices to show that the classifying map for the normal bundle lifts to BO (8m + 7) . Let A be the Steenrod algebra. Following the method of Gitler-Mahowald we begin by computing a minimal resolution of the A-module M = H^{j} (V_{8m+7}; ℤ_{2}) for j ≤ 8m + 10 (the top nonzero grading of H^{∗} (G_{2,4m+5}; ℤ_{2})). In this range the resolution is

M ⟵ C_{0} ⟵ C_{1} ⟵ ⋯

where M has a generator m_{7} in H^{8m+7} (V_{8m+7}; ℤ_{2}) , C_{0} is a free A-module on a generator g_{7} in grading 8m + 7 mapping into m_{7} and C_{1} is a free A-module on a generator h_{10} in grading 8m + 10 which maps into Sq^{2}Sq^{1}g_{7}. From this information we obtain the modified Postnikov tower

[figure: G_{2,4m+5}; ν; BO; w_{8m+8}; K (Z_{2}, 8m + 8); E_{1}; BO(8m + 7); k_{8m+10}; K (Z_{2}, 8m + 10)]

and it is our goal to lift the map ν up to BO (8m + 7) .

To see that it lifts to E_{1} we notice that the map labeled w_{8m+8} is the classifying map for the cohomology class w_{8m+8} ∈ H^{8m+8} (BO; ℤ_{2}) . But we have shown in the proof of Theorem 2.4 that w (ν) = 1 + w_{1} + w_{2} + w_{1}^{2} + w_{1}^{3} + w_{2}w_{1}^{2} thus ν^{∗} (w_{8m+8}) = 0, so the map w_{8m+8}^{∗} ∘ ν^{∗} is zero. Thus we have a lifting l : G_{2,4m+5} → E_{1}.

To prove that we can lift this map to BO (8m + 7) it suffices to show that l^{∗} (k_{8m+10}) is zero. But the lifting l is not unique. We can vary l^{∗} (k_{8m+10}) through H^{8m+7} (G_{2,4m+5}; ℤ_{2}) via the relation that produces k_{8m+10}, namely (Sq^{2} + w_{2} + w_{1}^{2}) Sq^{1}w_{8m+8} = 0 in H^{∗} (BO; ℤ_{2}) (which can be verified by the Wu formula). Thus it suffices to show that

l^{∗} (k_{8m+10}) = (Sq^{2} + w_{2} (ν) + w_{1} (ν)^{2}) Sq^{1}x

for some x ∈ H^{8m+7} (G_{2,4m+5}; ℤ_{2}) . But since w (ν) = 1 + w_{1} + w_{2} + w_{1}^{2} + w_{1}^{3} + w_{2}w_{1}^{2} we see that w_{1} (ν) = w_{1} and w_{2} (ν) = w_{2} + w_{1}^{2}. Thus we want to show l^{∗} (k_{8m+10}) = (Sq^{2} + w_{2}) Sq^{1}x for some x ∈ H^{8m+7} (G_{2,4m+5}; ℤ_{2}) .

The top nonzero class in H^{∗} (G_{2,4m+5}; ℤ_{2}) is in grading 8m + 10. In the description of cohomology we found in Corollary 2.2, this top class is w_{2}^{2^{i−1}−2}w_{1}^{2^{i}−2}. So either l^{∗} (k_{8m+10}) = 0 or else l^{∗} (k_{8m+10}) = w_{2}^{2^{i−1}−2}w_{1}^{2^{i}−2}. By Lemma 4.6, we see that w_{2}^{2^{i−1}−2}w_{1}^{2^{i}−2} = Sq^{2}Sq^{1} (w_{2}^{2^{i−1}−2}w_{1}^{2^{i}−5}) . Also by Corollary 2.2

w_{2}Sq^{1} (w_{2}^{2^{i−1}−2}w_{1}^{2^{i}−5}) = w_{2} (w_{2}^{2^{i−1}−2}w_{1}^{2^{i}−4})

= (w_{2}^{2^{i−1}−1}) w_{1}^{2^{i}−4}

= (∑_{j=0}^{i−2} w_{2}^{2^{j}−1}w_{1}^{2^{i}−2^{j+1}}) w_{1}^{2^{i}−4}

= (w_{1}^{2^{i}−1}) ∑_{j=0}^{i−2} w_{2}^{2^{j}−1}w_{1}^{2^{i}−2^{j+1}−3}

= 0 · ∑_{j=0}^{i−2} w_{2}^{2^{j}−1}w_{1}^{2^{i}−2^{j+1}−3}

= 0.

Hence, (Sq^{2} + w_{2}) Sq^{1} · (w_{2}^{2^{i−1}−2}w_{1}^{2^{i}−5}) = w_{2}^{2^{i−1}−2}w_{1}^{2^{i}−2} as required. Thus we can lift to BO (8m + 7) which completes the proof. □

5. Acknowledgement

Thanks to Don Davis for introducing me to the theory of modified Postnikov towers and for helpful comments on the initial draft of this paper. I am also grateful to the anonymous referees and editor who made helpful comments on the initial version of this paper.

References

[1] Borel, A.; La cohomologie mod 2 certains espaces homogènes, Comm. Math. Helv., 27 (1953), 165-197 [2] Chern, S. S.;On the multiplication in the characteristic ring of a sphere bundle, Ann. of Math.,

49 (1948), 362-372

[3] Cohen, R.; The immersion conjecture for differentiable manifolds, Ann. of Math. (2) 22 (1985), no. 2, 237-328

[4] Cox, D., Little, J., O’Shea, D.; Ideals, Varieties, and Algorithms, Springer-Verlag, 1992, ISBN: 0-387-97847-X

[5] Gitler, S., Mahowald, M.; The geometric dimension of real stable vector bundles, Bol. Soc. Mat. Mex., 11 (1966), 85-107

[6] Hiller, H., Stong. R. E.; Immersion Dimension for Real Grassmannians, Math. Ann., 255 (1981), no. 3, 361-367

[7] Lam, K. Y.; A formula for the tangent bundle of flag manifolds and related manifolds, Trans. Am. Math. Soc., 213 (1975), 305-314

[8] Lucas, E.; Théorie des functions numériques simplement periodiques, Amer. J. Math., 1 (1878), 184-240, 289-321

[9] Markl, M.; Lower Bounds for the embedding dimension of real oriented and nonoriented Grassmannians, Glas. Mat. Ser. III, 23 (43) (1988), no. 1, 213-219

[10] Milnor, J.,Stasheff, J.; Characteristic Classes, Princeton University Press, Annals of Mathematics Studies Number 76, 1974, ISBN: 0-691-08122-0

[11] Nussbaum, F.; Obstruction theory of possibly nonorientable fibrations, Northwestern Unversity PhD thesis (1970)

[12] Oproui, V.; Some non-embedding theorems for Grassmann manifolds G_{2,n} and G_{3,n}, Proceed. Edinburgh Math. Soc., 20 (1976-77), 177-185

[13] Oproui, V.; Some results concerning the nonembedding codimension of Grassmann manifolds in Euclidean spaces, Rev. Roumaine Math. Pures Appl., 26 (1981), no. 2, 275-286

[14] Whitney, H.; The singularities of a smooth n-manifold in (2n − 1) space, Ann. of Math. 45 (1944), 247-293

[15] Tang, Zi-Zhou; Non-immersion Theorems for the Grassmann Manifolds G̃_{2,n} and G_{2,n}, Chinese Sci. Bull. 38 (1993), no. 5, 353-355

Department of Mathematics, University of Scranton, Scranton, PA 18510

  E-mail address: monks@scranton.edu