Special embeddings into Euclidean space Svetlana I. Bogataya Higher school of economics

1. Special embeddings into Euclidean space Svetlana I. Bogataya Higher school of economics National research university; Moscow svetbog@mail.ru

2. X - compact space, n=dim X m=dim Rm C(X, Rm) – the space of all continuous maps: f, g : X Rm,

3. Gm,d – Grassman manifold ≡ the space of all d-dimensional subspaces in a fixed m-dimensional vector space Vm.

4. For g : X Rm let

5. 2× ?×2 problems • For a given n-dimensional (for all) space X estimate the minimal m such that the set of all “nice” maps g : X Rm is nonempty [is dense] in C(X, Rm) . • What means “nice” ?

6. Conjecture. For an n-dimensional compactum X the set of all maps g : X Rm such that dim Bq(g) ≤ qn-(q-d-1)(m-d) is dense (contains dense Gδ-set) in the space C(X, Rm).

7. The case when qn-(q-d-1)(m-d) ≤ -1, i.e. Bq(g) is empty was discussed in the report of S.Bogatyy. The case d=0 was proved by W.Hurewicz in 1933. Tuncali and Valov proved parametric version of Conjecture for d=0. Now we assume that 0 ≤ qn- (q-d-1)(m-d), 1 ≤d.

8. Theorem Bogataya- Bogatyy -Valov. The Conjecture is true in the following cases: • q=1; • q=2; • m ≥ qn; • d = q-2, m ≥ (q-1)n; • d=1, q=4, m=2n.

9. Let us formulate the geometric background of the Conjecture. For a fixedm-dimensional vector space Vm, subspace Wn and numbers d, r let us put For a fixedm-dimensional Euclidean space Rm, plane πn and numbers d, r let us put

10. Conjecture 2. For any subspaces and any ε>0 there exist subspaces W1n,…, Wqn such that ρ(Vin, Win)< ε and

11. Conjecture 3. For any subspaces and any ε>0 there exist subspaces such that and

12. Conjecture 4. For any planes and any ε>0 there exist planes such that and

13. Conjecture 5. Let points be such that all their coordinates are algebraically independent. Then

14. Theorem 1. If n+d ≤ m+r, r ≤ d, r ≤n, then

15. Theorem 3. If subspaces constitute direct sum and ni+d ≤ m+ri, r1+…+ rq≤ d, ri ≤ni, i=1,…,q, then

16. Theorem 4. Let the space be the sum of subspaces and the sum of any (q-1) subspaces is direct. If r ≤ni, i=1,…,q, then

17. Theorem 5(n=1). Let points be such that all their coordinates are algebraically independent. Then there are no more than 2 lines, which intersect 4 lines (A1, A2), (A3, A4), (A5, A6), (A7, A8).

18. Theorem 5. Let A: Vn  Vn be a linear operator of a vector space Vn. Let If all real eigenvalues of the operator A have multiplicity 1, then there are no more than n subspaces V2 such that has no more than n points.