Using the Borsuk-Ulam theorem (Jiri Matousek)

Errata and remarks

Concerning the second printing (2007)

• Page 25, the implication (LS-o) => (LS-c), an argument simpler than both of the given ones is this: Since F_i and -F_i do not intersect, then have at least some positive distance epsilon, and U_i can be defined as the (epsilon/4)-neighborhood of F_i.
• Exercise 2.1.8 should be deleted.
• Page 91, notes on nonembeddability theorems, the real projective space of dimension d is non-embeddable in R^{2d-1} ONLY for d a power of 2. (Noted by Peter Landweber.) See, for example, a survey by Skopenkov for additional information.
• Example (k) in Section 6.1, the indices of x in the image should be \pi^{-1}(1),\pi^{-1}(2), etc. (Noted by Peter Landweber.)
• Page 155 (143 in 1st printing), "homomorphisms" should be "homeomorphisms". (Noted by Peter Landweber.)
• Page 156, if there is a G-map X->Y, then Ind_G(X) CONTAINS Ind_G(Y), not the other way around as mistakenly stated. (Noted by Cyril Becker.)
• Page 168, last line, i=2 should be k=2. (Noted by Peter Landweber.)
• Theorem 6.8.1 (colored Tverberg), it should be said that the sets A_i have size d+1 (or that they are contained in the union of the C_j). (Noted by Peter Landweber.)
• Reference [Hae82] (Haefliger) should be [Hae62]; the paper is from 1962, not 1982. (Noted by Peter Landweber.)
• Reference [vK32], Van Kampen's initials should be E.R. van Kampen, not R.E. (Noted by Peter Landweber.)

These concern the FIRST PRINTING from 2003 ONLY, fixed in the second printing from 2007

• Page 5 top, the mapping F should go into X (noted by Jose Raul Gonzalez Alonso).
• Page 7 bottom, the index k is shifted by one. (Noted by Nati Linial.)
• Page 27, a nice combinatorial-geometric application of the Borsuk-Ulam theorem, which I forgot to mention, is a proof that any simplicial centrally symmetric d-dimensional convex polytope has at least 2^d facets, by Barany and Lovasz [Acta Math. Acad. Sci. Hungar. 40:323-329, 1982], also explained in Barany's survey [Bar93].
• Page 56, last sentence of the proof of the necklace theorem from the continuous version, sometimes both cuts need to be moved in the same direction along the necklace. (Noted by Gabor Tardos.)
• Page 61, exercise 3.3.2, there is a spurious "i" in the formula for the cycle length. (Noted by Haran Pilpel.)
• Page 74 bottom, ||K_1*K_2|| homeomorphic to ||L_1*L_2|| should be ||K_1*L_1|| homeomorphic to ||K_2*L_2||. (Noted by Michal Jablonowski)
• Page 81, the nerve theorem (Theorem 4.4.4.), an assumption is missing: The K_i together cover K, that is, every simplex of K is in some K_i. Similarly in the other version of the nerve theorem below. (Noted by Ben Braun.)
• Page 91 middle, Van Kampen's result actually guarantees only a PIECEWISE linear embedding of every n-dimensional pseudomanifold in R^{2n}. That is, some iterated subdivision can be embedded linearly. Linear embeddability doesn't seem to be known. (Noted by Lars Schewe.)
• Page 100, Gabor Tardos provided a simpler example of a non-tidy Z_2-space: Take a 2-sphere with small handles attached symmetrically near the north and south poles (homeomorphic to a "torus with two holes") with the antipodal Z_2-action. Then there is no Z_2-map of S^2 into it, but also no Z_2-map of it into S^1. Both of these facts can be shown by elementary arguments combined with the Borsuk-Ulam theorem.
• Page 112, first sentence of the paragraph "The Bier spheres", we associate a sphere with every simplicial complex on $n$ vertices EXCEPT for the (n-1)-simplex. (Noted by Gabor Tardos.)
• Page 112-114, Mark de Longueville ("Bier spheres and barycentric subdivision", manuscript, 2003, to appear in J. Combin. Theory Ser. A) discovered a shorter and more direct proof of the fact that Bier spheres are spheres.
• Page 115, many nonpolytopal triangulations, one should assume that n is at least 4, otherwise the considered Bier spheres may have fewer than 2n vertices. (Noted by Gabor Tardos.)
• Page 145, the suggested definition of n-fold k-wise deleted join of a space, at a closer look, doesn't seem very suitable unless k=n (which is the only case for which it is actually used in the book). Suggestions for better alternatives are most welcome. (Noted by Robert Vollmert.)
• Page 146, the first sentence of the paragraph "Free actions" is incorrect: The S_n-action is free on 2-wise deleted products, but not necessarily on 2-wise deleted JOINS. (Noted by Robert Vollmert.)
• Page 149, in the paragraph above Theorem 6.4.1, q(d-1) should be d(q-1). (Noted by Robert Vollmert.)
• Page 150 above the picture, i=2 should be k=2. (Noted by Peter Landweber.)
• In Example 6.7.4 on page 159, I don't know whether the claim about 3 intersecting triangles in R^3 is true with 9 points, but the suggested proof method works only for 11 points; this is the number given by the formula in Exercise 4. (Noted by Uli Wagner, ETH Zurich)