Algebraic topology: homology and cohomology by Andrew H. Wallace

By Andrew H. Wallace

This self-contained textual content is appropriate for complex undergraduate and graduate scholars and should be used both after or at the same time with classes typically topology and algebra. It surveys numerous algebraic invariants: the elemental workforce, singular and Cech homology teams, and quite a few cohomology groups.
Proceeding from the view of topology as a kind of geometry, Wallace emphasizes geometrical motivations and interpretations. as soon as past the singular homology teams, notwithstanding, the writer advances an figuring out of the subject's algebraic styles, leaving geometry apart so one can learn those styles as natural algebra. various workouts look in the course of the textual content. as well as constructing scholars' considering by way of algebraic topology, the workouts additionally unify the textual content, due to the fact a lot of them characteristic effects that seem in later expositions. vast appendixes provide useful reports of historical past material.

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We have f(Wk+l(~,B,u~)) = E(u~). Furthermore, if = ~, (f(Wk+l(m,B))) if in addition we obtain (when reducing mod 2 ) 2 = det we can w r i t e (Wb_k_i+j(B-m))l~i,j~a_k; a-k = 2a' and b-k = 2b', f ( ~ k + l ( m , ~ ) ) z det (£b - i + j ( ~ ' m ) ) i y i , j ~ a ' modulo the 2 - t o r s i o n of H4a'b' M; ~). Here the t o t a l S t i e f e l - Whitney or P o n t r j a g i n class o f the v i r t u a l bundle 6 - m is defined as the obvious q u o t i e n t ) . Moreover, k-morphism, from of the u let M ~(S) Thom c l a s s (formed we have i n of with is closed denote and the a tubular respect to u : ~ ~ ~ cohomology neighborhood suitable is class of a nondegenerate on the canonical M derived singularity orientations) S Then H(a-k)(b-k)~;~2).

We h a v e o o i , = tp (up to composition w i t h some i n v o l u t i o n on the t a r g e t group). Hence an isomorphism f o r i, n < 2(b-a). The second statement recovers in part a r e s u l t of I . ( [z~], 8 . 1 ) . where James is 53 Given an element in Proof. l(Pb'l/pb-a-1), ~ pb-1/pb-a-1 and transverse consider the morphism u : Dn x which is smooth in a neighborhood pa-1 to ~a to a l i n e a r homotopy between i o f , inclusion ~a¢ ~b-a x ~a, S(½) To compute o o i , [ f ] Dn x ~b, given at which corresponds ~Dn = Sn ' l , given at the center of degenerate s i n g u l a r i t y of the sphere we may represent i t u of radius l i e s at the set ½, and Dn.

2 to) our s i n g u l a r i t y invariants, and c h a r a c t e r i s t i c and c l a s s i c a l classes. of § 2, and f o r s i m p l i c i t y Write morphism r f for indicate the connection between cohomology o b s t r u c t i o n s Thus we place o u r s e l v e s in the s i t u a t i o n we assume t h a t (a-k)(b-k) M is compact. ~H~ f ~ Hr(M,@M;~r_I ) E(u ~) Here we are dealing with singular (co)homology with twisted coefficients. g. ~@ is associated with the orientation line bundle @ (or, equivalently, with (a-k)(Wl(~)+Wl(~))+(a+b)Wl(~)+Wl(M)).

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