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12. Prove that D is homeomorphic to the hemisphere of the dimension σk+1 − k − 1. Thus D is a closed cell of dimension σk+1 −k −1. 2. We define the map f : E(σ1 , . . , σk ) × D −→ E(σ1 , . . , σk , σk+1 ) by the formula f ((v1 , . . , vk ), u) = (v1 , . . , vk , T u) where T is given by (13). We notice that vi , T u = T ǫi , T u = ǫi , u = 0, i = 1, . . , k, and T u, T u = u, u = 1 by definition of T and since T ∈ O(n). 13. Recall that σk < σk+1 . Prove that T u ∈ H σk+1 if u ∈ D . NOTES ON THE COURSE “ALGEBRAIC TOPOLOGY” 31 The inverse map f −1 : E(σ1 , .

Then there exists a cell map g : X −→ Y such that g|A = f |A and, moreover, f ∼ g rel A. First of all, we should explain the notation f ∼ g rel A which we are using. Assume that we are given two maps f, g : X −→ Y such that f |A = g|A . A notation f ∼ g rel A means that there exists a homotopy ht : X −→ Y such that ht (a) does not depend on t for any a ∈ A. Certainly f ∼ g rel A implies f ∼ g , but f ∼ g does not imply f ∼ g rel A. 3. Give an example of a map f : [0, 1] −→ S 1 which is homotopic to a constant map, and, at the same time f is not homotopic to a constant map relatively to A = {0}∪{1} ⊂ I.

Is ) as above. The Young tableaus were invented in the representation theory of the symmetric group Sn . This is not an accident, it turns out that there is a deep relationship between the Grasmannian manifolds and the representation theory of the symmetric groups. NOTES ON THE COURSE “ALGEBRAIC TOPOLOGY” 33 5. 1. Borsuk’s Theorem on extension of homotopy. We call a pair (of topological spaces) (X, A) a Borsuk pair, if for any map F : X −→ Y a homotopy ft : A −→ Y , 0 ≤ t ≤ 1, such that f0 = F |A may be extended up to homotopy Ft : X −→ Y , 0 ≤ t ≤ 1, such that Ft |A = ft and F0 = F .