By Hans-Joachim Baues

The writer, a number one determine in algebraic topology, presents a latest therapy of an extended verified set of questions during this vital learn sector. The book's significant objective--and major result--is the class theorem on k-variants and boundary invariants, which complement the classical photo of homology and homotopy teams, besides computations of varieties which are got by means of employing this theorem. examine mathematicians in algebraic topology should be attracted to this new try and classify homotopy different types of easily hooked up CW-complexes.

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**Extra resources for Homotopy Type and Homology**

**Sample text**

Whitehead with the homotopy groups of a certain space FX. This is related to results of Dold and Thom on the infinite symmetric product of X and to results of Kan on the loop group of X. The results here are useful background knowledge on the r-groups and on Whitehead's exact sequence. 2. 5 is an essential step for the proof of the boundary classification theorem below; a definition of b' as given here by a fibre sequence is not appropriate for this proof. Let X be a simply connected CW-complex with base point and let SPX be the infinite symmetric product of X.

Now we define the boundary operator bn above by the formula bn(a)=cpQ{ }+0(bn+1)(20) Here we use { ) in (16) and (b,, 1) * : Ext(A, H,, Ext(A, r,,). The element (20) is well defined. 4. Similarly we see that b' (a) depends only on the homotopy class (a) of the chain map a. 6) Lemma The boundary operator bA is natural in X. Let w:W-Yrt-1 be chosen for Y (with C. 5 is chosen for X. 12 in Baues [AH], is a twisted map since n z 3. 12 in Baues [AH]. Here 710 is the restriction of the cellular map F and fo is a map with the following properties.

5 is chosen for X. 12 in Baues [AH], is a twisted map since n z 3. 12 in Baues [AH]. Here 710 is the restriction of the cellular map F and fo is a map with the following properties. Let q:M(H,,,n-1)=Cd- M(Bn,n) 2 INVARIANTS OF HOMOTOPY TYPES 50 be the quotient map. Clearly, q* induces the inclusion 0 in the universal coefficient sequence. Now the restriction (2) f= = S01M(H,,,n-1) is the sum of the following four maps: 61:M(H",n - 1) -q M(B",n) ->M(Z;,+l,n) cW 2:M(H",n-1)--M(H,,n-1)cW 63:M(H",n-1) *M(B",n)->M(Bn_1,n-1)cW Y2.