# Algebra, Algebraic Topology and their Interactions: by Jan-Erik Roos By Jan-Erik Roos

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Additional resources for Algebra, Algebraic Topology and their Interactions: Proceedings of a Conference held in Stockholm, Aug. 3 - 13, 1983, and later developments

Example text

Suppose (T 3 ..... y' = G = L ~ B ~ R . tTitT2 .. tTl when~-- (T1.. t. all have a common ij y by deleting this factor. R = k @ RI B where relation terms ~: R + @ L + ---+ Now, put be of degree , we shall adjoin new variables to IYl - I . ,t p) Y~ B denote the image of B/ = B . To obtain the promised algebra , along with relations with = N > 0 . Let y'~ G as • = y and ~l<~,y,> under the map and define I = G , we have oI<~'> G---+ G . Hence D(z) ! O(z) = (I + p2z)(1 + z)B(z) . 51 Claim: co(D) !

R-l(b) . Its hemotopy type does is any continuous map then the homotopy fibre of ~ and (any) inclusion and is an extremely to a fibration The analogue for to an augmentation E-~B h-DGF, h-DGF~ s R ~--~k morphism homotopy fibre of {b} ~ B. It depends only on important invariant. A representative and take a fibre, ~-l(b) into a fibration P ~ B h-DGF~ (cf. 12) of ~ . As in topology it is an extremely less attention, presumably because even when A representative and for this homotopy fibre can be constructed R -~ S of #; then ~ Choose a free model R~ of E; then S represents S invariant.

Corollary Suppose L, R trivial multiplication ~: R + @ L + ~ A of the zero map of a map and on R L+ Moreover, @ A if L @ A ~ and L ~ A L @ A @ R L @ A @ R @ ~ I ~A ~ A @ A-----~ A c A L and R have is an algebra by means is an algebra by means defined as zero on R + @ (k @ A +) as the composition GL, G R and G A respectively and are algebras such that of graded vector spaces. Then R + ~ (L @ A) + ~ A (L+) 2 = (R+) 2 = 0). e. ~ L @ A ~ R . are minimal generating sets for has the presentation k/<~ > , then L, R and A L @ A ~ R has the presentation k < G L U G A Y GR>/<~ U {al, ii', rr', ra, rl - @(rSl); i,i'~ GL,aeGA,r,r'eGR}~ Proof A trivial application of Proposition 2 shows that L 8 A is an algebra with presentation k / < ~ U Moreover R (ii', al ; I,I' C G L , a C GA}>.