# Quail Entry: Homology Basics

This happened probably closer to mid December, but I’m posting it now

### Simplicial and Singular

What a simplicial is roughly captured in the fact that the figure below is a simplicial complex

but if the edges were oriented cyclically then you would not have a simplicial complex (would fail order preserving). The order preserving bit is probably so that is zero. Following Hatcher denotes simplicial homology.

Have isomorphism

Natural map

1. First do case ,

use induction, l.e.s for

show directly , this finish with 5-lemma

2. For show map is inj and surj via

image of chain is compact and hence contained in some ; then apply 1.

3. For arb use l.e.s for .

pf provides motivation for relative homology groups

Via the above isomorphism, have easy properties

(1) groups are fin. gen.

(2) for

(easy to see with )

(3) is homemorphic

In fact, enough hmtpy equivalent. This follow from a more general statement about

hmtpy and homology: if then

can be used to get

becomes

now check

simple to see if is hmtpy equivalence get desired iso

Relative Homology Groups

Any gives rise to s.e.s of complexes

and general homological algebras says there is long exact seqence of homology, the third group that appears is labeled so get

is a good pair if is closed and there is open nbd. of that deformation retracts to . This definition is made because of

Good Pair Isomorphism:

This is proved via

Excision: with then gives iso

proved via crazy technical baryocentric subdivision stuff…

and

l.e.s of a triple which arises from the s.e.s of complexes

for a good pair have and deformation retracts to so l.e.s is

and middle map is iso because .

and diagram

where , the middle doesn’t matter, all that is left to show is that the right is an iso, and this follows essentially because the part where its not an iso is excised.

Disjoint union corollary: a good pair, get

And for arbitrary pair have a series of isos

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- Published:
- January 2, 2010 / 1:13 pm

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- Quail, wall scribble

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