# Differential Forms in Algebraic Topology by Raoul Bott

By Raoul Bott

The tenet during this publication is to take advantage of differential varieties as an relief in exploring a number of the much less digestible facets of algebraic topology. Accord ingly, we circulate essentially within the realm of soft manifolds and use the de Rham concept as a prototype of all of cohomology. For functions to homotopy thought we additionally talk about when it comes to analogy cohomology with arbitrary coefficients. even supposing now we have in brain an viewers with past 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 beneficial, yet not likely worthwhile. in the textual content itself now we have acknowledged with care the extra complicated effects which are wanted, in order that a mathematically mature reader who accepts those heritage fabrics on religion can be capable of learn the whole booklet with the minimum necessities. There are extra fabrics right here than should be quite coated in a one-semester direction. yes sections might be passed over before everything studying with out lack of continuity. we now have indicated those within the schematic diagram that follows. This ebook isn't meant to be foundational; quite, it is just intended to open many of the doorways to the ambitious edifice of recent 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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By hypothesis Poincare duality holds for U o u ... U Up-it U" and (U o u ... u U p- 1) n Up, so it holds for U 0 u ... u U ,-1 U Up as well. This induction argument proves Poincare duality for any orientable manifold having a finite 0 good cover. ' The finiteness assumption on the good cover is in fact not necessary. By a closer analysis of the topology of a manifoh:l, the MayerVietoris argument above can be extended to any orientable manifold (Greub, Halperin, and Vanstone [1, p. 198 and p. 14]).

By the same argument as for the degree of a proper map between two Euclidean spaces, the degree of a map between two compact oriented manifolds is an integer and is equal to the number of points, counted with multiplicity ±'1, in the inverse image of any regular point in N. 9) = H*(M) ® H*(F). This means Hn(MXF)= EB HP(M)fiJHq(F) foreverynonnegativeintegern. p+q-n More generally we are interested in the cohomology of a fiber bundle. Definition. Let G be a topological group which acts effectively on a space F on the left.

J: :~ , + (-l)'-l(1t*q,{dx(J: ~) + f dt]' §4 35 Poincare Lemmas In either case, 1 - 1t* 0 s* = (-1)'-1(dK - Kd) on Q4(R" x IR). This proves 'I. ,. 1. The maps H*(IR" x R 1) ~ H*(R") are isomorphisms. By induction, we obtain the cohomology of R". 1 (Poincare Lemma). H*(A") = H*(point) = { 0R in dimension 0 elsewhere. Consider more generally M 7t X R1 IIMs . If {U«} is an atlas for M, then {U cc x R I } is an atlas for M x R 1• Again every form on M x RI is a linear combination of the two types of forms (I) and (II).