# 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

This requires path connected.

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

show constant paths are boundaries and for paths that 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

.

The way to see it is

Construct a surjection ; the map just restricts cocycles to be evaluated on cycles; can show surj by const. lift

get

rederive this s.e.s by looking at [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 and just reappear becasue boundry maps are zero)

note in l.e.s is just the dualization of the inclusion

split up into s.e.s again to get

Sequence of interest is ; is free resolution, dual and use derived functor to get

and

Dualize to get univ. coeff. for homology:

.

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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- January 1, 2010 / 11:50 pm

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