Self-Maps of Flag Manifolds

Abstract

Abstract.Rationally, a map between flag manifolds is seen to be determined up to homotopy by the homomorphism it induces on cohomology. Two algebraic results for cohomology endomorphisms then serve (a) to determine those flag manifolds which have (nontrivial) self-maps that factor through a complex projective space, and (b) for a special class of flag manifolds, to classify the self-maps of their rationalizations up to homotopy. 0. Introduction. The homotopy classes of self-maps of a complex projective space are classified by an integral degree, and we show that, for rational homotopy, there is an analogous result for a much larger class of homogeneous spaces. These spaces are quotients of a unitary group by a closed connected subgroup of maximal rank, and, just as for projective spaces, the degree of a self-map is determined by its effect on a two-dimensional cohomology class. The first step is to observe that homotopy classes of endomorphisms of the minimal model for the rational homotopy type of such a space correspond bijectively with endomorphisms of the rational cohomology algebra (see Theorem 1.1). The second step is to classify the cohomology endomorphisms. Such a classification was given in [GH1] for the Grassmann manifold of complex p-planes in complex /i-space (for n greater than 2p2), and it follows that the rational homotopy classes of self-maps are indeed classified by a rational degree.Here, in Theorems 1.3 and 1.4, we give the analogous result for the flag manifolds M(l,p, q)=U(l+p + q)/(U(l) X U(p) X U(q)).The cohomology endomorphisms fall into a finite number (often one) of families, each of which is classified by a rational degree. Some of these families are represented by the endomorphisms induced by conjugation in U(l + p + q) by elements of the normalizer of U(l) X U(p) X U(q), but if 1, p, and q are distinct we get only the identity in this way. The rest are represented by idempotents, which we call projective endomorphisms because they factor through the cohomology of complex projective space. Such projective endomorphisms (for any flag manifold) are in turn classified in Theorem 1.3 by factorizations over Z of the polynomial 1 -/".

Fixed points on flag manifolds

“…(Note that this corrects the proof of Theorem 5 in [3], where p 2 was inadvertently omitted from the formula for Ad (ii): If n x -2 or 3, then H^(RM(n lf n 2 , n 3 ); Q) is isomorphic (with a shift in grading) to JΪ*(CΛf(l, n 2 , ή 3 ); Q). We proceed as in the proof of (i), with Theorem 1.4 of [4] restricting the possibilities for cohomology endomorphisms and a similar 0* 1 argument to show that the projective endomorphism of degree -1 (which has Lefschetz number 0 if ΰ 2 + n z is odd) is not realized by a self map.…”

“…This is well known to be true for complex projective spaces (k = 2 and n ι = 1) and has been proved for many Grassmann manifolds (fc = 2 and either n t ^ 3 or n 2 2> 2n\ -n t -1 [5,3]). Here we verify the following additional cases.…”

“…By Theorem 1 of [3] (with the slight improvement noted in [4]), H* (CM(n 2 , ή 3 ); Q) admits only grading endomorphisms when either n 2 i^3 or n z ^ 2n\ -2n 2 -1, which are equivalent to the inequalities in the statement of (i). The grading endomorphism of degree -1 has Lefschetz number 0 when n 2 and n B are both odd.…”

Suspension Splittings and Self-maps of Flag Manifolds

“…Note that Tel(c 5 ) ≃ A (5,9) and Tel(c 4 ) is a three-cell complex whose 9-skeleton is A (5,9). We claim that Tel(c 4 ) ≃ A (5,9) ∨ S 13 . To see this we show that the attaching map g : S 12 −→ A (5,9) for the top cell is null homotopic.…”

“…There has been considerable interest in trying to determine the homotopy classes of the self-maps of the quotient space G/T. One method commonly adopted in [6,9,19,26] is to study the image of the map r : [G/T, G/T] −→ Hom alg (H * (G/T), H * (G/T)).…”