By Seebach J.A., Steen L.A.

This compendium includes major examples of topological areas, every one analyzed intimately. Numbering nearly one hundred fifty, the examples variety from the conventional to the vague and are preceded by means of a succinct exposition of normal topology and easy terminology and idea. Oveer 25 Venn diagrams and reference charts summarize the examples` houses and make allowance the reader to test fast for examples with prescribed houses.

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Id, g, id, . . , id), i = 1, . . , m, for f ∈ O(m), g ∈ O(n). , [KSV96]. 5 (The Riemann surface and the endomorphism operads). , n inputs and 1 output. Another example is the endomorphism operad of a vector space V : End V (n) = Hom(V ⊗n , V ), the space of n-linear mappings from V to V . 4. An algebra over an operad O (in other terminology, a representation of an operad ) is a morphism of operads O → End V , that is, a collection of maps for n ≥ 0 O(n) → End V (n) compatible with the symmetric group action, the unit elements, and the compositions.

Cohen is very interesting. 1 (F. Cohen [Coh76]). , a graded vector space V with a unit element e, a dot product ab, and a bracket [a, b] deﬁned, so that the dot product deﬁnes the structure of a graded commutative associative unital algebra, the bracket deﬁnes the structure of a graded Lie algebra on the suspension V [−1], which is the same as V but with a grading shifted by −1, and the bracket is a degree-one derivation of the dot product: [a, bc] = [a, b]c + (−1)(|a|+1)|b| b[a, c] for all a, b, and c ∈ V .

2 The cacti operad The construction and results in this section have been announced in [Vor01]. The BV structure arising in string topology at the level of homology comes from an action of a cacti operad C at the motivic level, quite close to the category of topological spaces. The kth component C(k) of the cacti operad C for k ≥ 1 may be described as follows. C(k) is the set of tree-like conﬁgurations of parameterized circles, called the 38 Chapter 2. The cacti operad 2 1 4 3 k 5 ... 10: A cactus lobes, labeled by numbers 1 through k, of varying (positive) radii, along with the following data: (1) the choice of a cyclic order of components at each intersection point and (2) the choice of a marked point on the whole conﬁguration along with the choice of one of the circles on which this point lies.