By Botvinnik B.

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**Example text**

Smaller simplices as follows. First let us look at the example: 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 11 00 11 00 00 11 11 00 00 11 11 00 11 00 11 00 00 11 11 00 00 11 11 00 00 11 Figure 17 11 00 00 11 NOTES ON THE COURSE “ALGEBRAIC TOPOLOGY” 39 In general, we can proceed by induction. The picture above shows a barycentric subdivision of the simplices ∆1 , and ∆2 . Assume by induction that we have defined a barycentric subdivision of the simplices ∆j for j ≤ q − 1.

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 .

We have to define an extension of F1 from the side g(S n ) × I and the bottom base g(Dn+1 ) × {0} to the cylinder g(Dn+1 ) × I . By definition of CW -complex, it is the same as to construct an extension of the map ψ = F (n) ◦ g : (Dn+1 × {0}) ∪ (S n × I) −→ Y to a map of the cylinder ψ ′ : Dn+1 × I −→ Y . Let η : Dn+1 × I −→ (Dn+1 × {0}) ∪ (S n × I) be a projection map of the cylinder Dn+1 × I from a point s which is near and a bit above of the top side Dn+1 × {1} of the cylinder Dn+1 × I , see the Figure below.