> > Algebraic topology. Errata (web draft, Nov. 2004) by Hatcher A.

# Algebraic topology. Errata (web draft, Nov. 2004) by Hatcher A.

By Hatcher A.

Similar geometry and topology books

Grassmannians and Gauss Maps in Piecewise-Linear and Piecewise-Differential Topology

The booklet explores the potential for extending the notions of "Grassmannian" and "Gauss map" to the PL type. they're special from "classifying area" and "classifying map" that are basically homotopy-theoretic notions. The analogs of Grassmannian and Gauss map outlined comprise geometric and combinatorial details.

Additional info for Algebraic topology. Errata (web draft, Nov. 2004)

Sample text

47) equals − β, [π (α), T (X)] + T ([π (α), X]) . 49) for v˜ constant; in this case, the equation is just the invariance of π . 48) equals (α) (ρ ∨ dJ (X)). 24 H. Bursztyn and M. 50) J )(X) . On the other hand, for all vector ﬁelds Y on M and g-valued 1-forms ν on G, we have iX LY (J ∗ ν) = LdJ (X) (ν(dJ (Y ))) + (dν)(dJ (Y ), dJ (X)). 47) and prove the claim. 41) evaluated at a vector ﬁeld X is b, dρ ∨ ((σ ∨ )∗ a, V ) − a, dρ ∨ ((σ ∨ )∗ b, V ) − a, LV (D)b . 52) where we have used again that dJ π = −((σ )∨ )∗ (ρM )∗ .

On the other hand, if a Lie algebroid is integrable, then there exists a canonical source-simply-connected integration G(A); see [14]. If M is a point, then a Lie groupoid over M is a Lie group, and the associated Lie algebroid is its Lie algebra. 1 (transformation Lie groupoids). Let G be a Lie group acting from the left on a manifold M. The associated transformation Lie groupoid , denoted by G M, is a Lie groupoid over M with underlying manifold G × M, source map s(g, x) = x, target map t(g, x) = g · x, and multiplication (g, x) · (g , x ) = (gg , x ).

In Poisson geometry, Poisson maps are always associated with Lie algebroid actions: If (Q, πQ ) and (P , πP ) are Poisson manifolds, then any Poisson map J : Q → P induces a Lie algebroid action of T ∗ P on Q by 1 (P ) −→ X (Q), α → πQ (J ∗ α). 16) When the target P is the dual of a Lie algebra, we recover a familiar example. 8 (inﬁnitesimal Hamiltonian actions). Consider g∗ equipped with its Lie– Poisson structure. 5) is that of a transformation Lie algebroid g g∗ with respect to the coadjoint action.