Fixed-point-free involution

Specifying F = Z2 in Definition 8.12, we see that a Z2-space X is a CW complex equipped with a fixed-point-free involution, that is, with a continuous map 7 X —> X such that 7 is an identity map. Correspondingly, we have Z2-maps as continuous maps that commute with these involutions, and a category of Z2-spaces as objects and Z2-maps as morphisms, which is called Z2-Sp. 

Let X be a regular CW complex and a Z2-space, and denote the fixed point free involution on X by 7. As mentioned above, one can choose a simplicial structure on X such that 7 is a simplicial map. We define a Z2-map p X ... 

Cobordism invariants/ the Kervaire invariant and fixed point free involutions... 

Fixed point free involutions on homotopy spheres Tohoku J. Math. 25, 69- 88 (1973) G.Brumfiel and J.Milgram... 

The crucial fact needed to see (19.12) is that the involution 7 is fixed-point-free. Indeed, for any nonnegative integer i, the vector space C (X Z2) has a basis consisting of 9(c ), where c ranges over a set of i-dimensional cells of X, obtained by choosing exactly one cell from each orbit of the Z2-action on the set of all i-dimensional cells of X. This shows that lm(0) = C (X Z2). On the other hand, equation (19.11) implies immediately that Im(q ) C Im( ). In fact, one sees further that q C XjZ2 l 2) —> Im(6 ) is an isomorphism. We let p lm(0) —> C (X/Z2 Z2) denote its inverse. 

Scroll compressors are currently used in relatively smaU-sized installations, predominantly for residential air-conditioning (up to 50 kW). They are recognized for low-noise operation. Two scrolls (free-standing, involute spirals bounded on one side by a flat plate) facing each other form a closed volume while one moves in a controlled orbit around a fixed point on the other, fixed scroll. 

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