By Raoul Bott

The guideline during this booklet is to exploit differential types as an reduction in exploring a number of the much less digestible points of algebraic topology. Accord ingly, we flow essentially within the realm of delicate manifolds and use the de Rham thought as a prototype of all of cohomology. For functions to homotopy concept we additionally talk about when it comes to analogy cohomology with arbitrary coefficients. even though we've in brain an viewers with previous publicity to algebraic or differential topology, for the main half an exceptional wisdom of linear algebra, complicated calculus, and point-set topology may still suffice. a few acquaintance with manifolds, simplicial complexes, singular homology and cohomology, and homotopy teams is useful, yet not likely helpful. in the textual content itself we have now acknowledged with care the extra complicated effects which are wanted, in order that a mathematically mature reader who accepts those historical past fabrics on religion can be in a position to learn the complete publication with the minimum must haves. There are extra fabrics the following than may be kind of coated in a one-semester direction. convinced sections might be passed over firstly examining with out lack of continuity. we've indicated those within the schematic diagram that follows. This ebook isn't meant to be foundational; quite, it's only intended to open the various doorways to the bold edifice of contemporary algebraic topology. we provide it within the desire that such an off-the-cuff account of the topic at a semi-introductory point fills a spot within the literature.

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**Extra resources for Differential Forms in Algebraic Topology**

**Sample text**

The set of open covers on a manifold is a directed set, since any two open covers always have a common refinement. A subset J of a directed set I is co/inal in I if for every i in I there is a j in J such that i

We will show that'" is an isomorphism. If M = R m, this is simply the Poincare lemma. In the following we will regard M x F as a product bundle over M. Let U and V be open sets in M and n a fixed integer. From the Mayer-Vietoris sequence • •• --+ H'(U u V) --+ H'(U) e H'(V) --+ H'(U n V)··· we get an exact sequence by tensoring with H"-'(F) ••• --+ H'(U u V) ® HPI-'(F) --+ (H'(U) ® H"-'(F) E9 (H'(V) ® HIt-P(F» --+ H'(U n V) ® HIt-'(F) --+ ••• since tensoring with a vector space preserves exactness.

H. R". H~(IR") ~ 0 in dimension n otherwise. , by inte- To determine a generator for H~(IR"), we start with the constant function 1 on a point and iterate with e•. This gives e(Xl) dXl e(X2) dX2 ... e(xII) dx,.. 40 I de Rham Theory So a generator for H:(jRlI) is a bump n-form ~(x) dx 1 ••• dx" with r cx(x) dXl ... dX = 1. n JR- The support of ~ can be made as small as we like. REMARK. This Poincare lemma shows that the compactly supported cohomology is not invariant under homotopy equivalence, although it is of course invariant under diffeomorphisms.