Quail Entry: abel of pi_1, univ. coeff thm and noted pages

This happened around 12/21/09.  At this point I had also pretty much finished reading chapter 2 of Hatcher ( homology), but a post of that will come later.

Abelianization of pi_1

The thm is that H_1 = \pi_1/[\pi_1, \pi_1]

This requires path connected.

pf by construct map \pi_1 \to H_1 since each loop is also a cycle (observe that this is a group homo)

show constant paths are boundaries and for  paths f,g that im(f \cdot g) - im(f) - im(g) is a boundary. etc.

show surjective by taking a sum of cycle and do some rearanging to get it as a sum of loops, show can lift each loop (might need to compose with path from pi-1 base point to base point of H_1 loop)

this is where path connetedness comes in.

Universal Coefficients Thm

Other thing I did was start cohomology and review universal coefficient thm for homology.  Its

H^n = \hom(H_n, G) \oplus Ext(H_{n-1}).

The way to see it is

Construct a surjection H^n \to \hom(H_n, G); the map just restricts cocycles to be evaluated on cycles; can show surj by const. lift

get 0 \to Ker \to H^n \to \hom(H_n, G) \to 0

rederive this s.e.s by looking at  Z^n \leftarrow C^n \leftarrow B^{n-1} [as s.e.s of complexes]

This is exact because the dual sequence is split exact (this uses that subgroup of a free group is free and can be proved via van kampfen)

Suggestion: determine how to show free group on two generator contains free group of any countable number of gen and of infinite index…

get long exact sequence of homology (but Z^n and B^n just reappear becasue boundry maps are zero)

note in l.e.s  j: Z^n \to B^n is just the dualization of the inclusion i : B_n \to Z_n

split up into s.e.s again to get  0 \to coker(j) \to H^n \to ker(j) \to 0

Sequence of interest is 0 \to B_n \to Z_n \to H_n; is free resolution, dual and use derived functor to get
\ker(j) = \hom(H_n,G) and coker(j) = Ext(H_{n-1}, G)

Dualize to get univ. coeff. for homology:

H_n(G) = H_n \otimes G \oplus Tor(H_{n-1},G).

Pages from Hatcher

44,45:
example showing the necessity of path connecteness of van kampen’s thm, and proof.

49,50:
Hawaiian earing, and
computing pi-1 by attaching maps of 2-cells…

75:
group actions on spheres and
calculation of fundamental group of RP^N

104,109:
Dunce cap and
rep. cycles roughly by their image

125:
Any H_n(X,A) can be rep by abso. homology of a space X \cup Cone(A)
—> uses. long. seq. of pair and excision.

143:
Calculation of homology groups for Klein and related spaces

175,176:
use:  odd map f: S^n –> S^n must have odd degree.
to prove borsuk-ulam:
any g: S^n —> R^n has x s.t.
g(x) = g(-x)

f(x) = g(x) – g(-x), apply prev. prop. want f(x) =0, if not can divide and normalize
to get contradictory homotopy…

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