By Harry Oliver
The attention-grabbing and humorous origins of daily objects-bliss for historical past hounds, language fans and minutiae buffs.
during this pleasant quantity, Harry Oliver unearths the main strange and unforeseen tales at the back of the family must haves, toys, universal gadgets, technological advances, and daily goods all of us take without any consideration. Who hasn't puzzled:
?Whether Thomas Crapper rather invented the bathroom
?What twist of fate ended in the discovery of the microwave
?Why it took approximately 20 years for somebody to ultimately choose to slice bread
?How laziness ended in the discovery of the dishwasher
?Which discovery made the milkshake attainable
?Which king's fancy for his mistress encouraged the 1st elevator
Read Online or Download Bubble Gum and Hula Hoops: The Origins of Objects in Our Everyday Lives PDF
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Extra info for Bubble Gum and Hula Hoops: The Origins of Objects in Our Everyday Lives
1 Theorem. Let G be cyclic. Then the canonical homo- LZ/m morphisms are isomorphisms. Proof. We first show that JO(G) of dimension functions. 4 whenever ~ \' d~k we obtain LZ/m is two-dimensional and one-dimensional otherwise. 3 Lemma. A function Dim kid LZ/d. dimV(d). u(d)n(LZ/kd). An irreducible real representation of if 2: C(G) n E C(G) is contained in the image of if and only if u(d)n(LZ/kd) _ 0 mod 2 'm/k I > 2. Proof of the Le~~. 2) and the remarks thereafter the con- gruences are necessary.
Then the following holds. 11 Theorem. X has the G-homotopy type of a finite complex if and only if for every element n(H) H ( G such that is a pt-1_ th power Proof. TOM DIECK . G, G/H is non-cyclic the mod pt = [G/HI. 31 5. Cyclic groups. 1 Theorem. Let G be cyclic. Then the canonical homo- LZ/m morphisms are isomorphisms. Proof. We first show that JO(G) of dimension functions. 4 whenever ~ \' d~k we obtain LZ/m is two-dimensional and one-dimensional otherwise. 3 Lemma. A function Dim kid LZ/d.
S n Thus if there exists a homotopy equivalence, then the stated congruence must hold. ) jO(G,f) 0 be the jO group of G-representations which are direct sums of faithful irreducible representations. :1 } , ~ j 0 (G , f) By lifting representations of m = I Gr. 9 Proposition. s = ~ jO(G). (cHIH ( G) jO(G/H,f) -7 jO(G) H~G is an isomorphism. bH:jO(G) -7 jO(G/H). Proof. Taking H--fixed paints defines a map bHc K = 0 jO(G/K,f) We have equals if H ~ K; -p jO(G/K) 7 and for H ( K jO(G/H). this composition Clearly s is surjec- tive.