M2 Midterm 2 Review

Part 1: Topology

Homotopy and fundamental group

(Munkres ch. 9, §51–55 + 58–60)

• Homotopy: for continuous, a homotopy is continuous with , .
• Paths are path-homotopic if there is a homotopy fixing end points. (Lec. 12)
• Path-homotopy classes of paths in form a groupoid for path composition ( from to , then from to ). (Lec. 13)
• Path-homotopy classes of loops based at (= paths ) form the fundamental group (product = composition, identity = constant loop, inverse = reverse loop). (Lec. 14)
• in the same path component of (by attaching a path: ).
• induces a homomorphism . Functorial: .
• Examples: , convex subsets of , and for are simply connected (). , , the free group . (Lec. 15–16)
• Applications of :
  • there is no retraction (i.e. continuous, );
  • every continuous has a fixed point () (Brouwer).
• Deformation retraction: a retraction () such that is homotopic to among maps that leave fixed, i.e. with Then (, inverse isomorphisms). (Lec. 13–14)
• The same holds more generally for homotopy equivalences , , .
• Covering spaces: such that every has a neighborhood evenly covered by ( = disjoint union of slices , each a homeomorphism). (Lec. 15)
• Every path starting at has a unique lift starting at . (Path) homotopies lift to (path) homotopies.
• Looking at end points of lifts of loops in , we get the lifting map .
• Those loops which lift to a loop in form a subgroup , and . (Lec. 16–17)
• A map lifts to iff .
• Classification of covering spaces (up to equivalence) classification of subgroups (up to conjugacy). (Munkres §79–80)
• Universal cover: simply connected (i.e. ).
• Van Kampen: , open, path-connected (Lec. 18; Munkres §70)
  • is generated by the images of and ;
  • if then is the free product ;
  • otherwise, it is the quotient of by the smallest normal subgroup that makes for all .

Part 2: Real analysis

1. Sequences and series

(Rudin ch. 3; McMullen §4; Lec. 19)

• In a metric space , iff , i.e. such that , .
• is a Cauchy sequence such that , .
• Convergent Cauchy; if is complete (e.g. ; with the sup norm; compact metric spaces; …).
• Compactness of every bounded sequence in has a convergent subsequence.
• A monotonic sequence in (e.g. ) converges iff it is bounded ().
• means such that ( converges in ).
• If is bounded then the largest limit of a convergent subsequence of . Similarly .
• A series converges iff the partial sums are a convergent sequence.
• converges . For , converges iff the partial sums are bounded.
• Comparison criterion: , convergent convergent.
• converges iff ; converges iff .
• converges absolutely if converges; absolutely convergent convergent, but not : e.g. alternating series (, ) always converge.
• Root test: converges (absolutely); diverges.

2. Continuous real functions of 1 variable

(Rudin ch. 4 and 7; McMullen §3; Lec. 19)

• Continuity at ( such that , ) . Infinite limits, limits at : work in .
• Compactness of continuous functions on are uniformly continuous, i.e. such that , (the same works for all ).
• Intermediate value theorem: is connected it contains all reals between and .
• Extreme value theorem: is compact it is bounded and contains its inf and sup.
• pointwise if , (= product topology).
• uniformly if .
• If the are continuous and uniformly then is continuous, i.e. is closed in the uniform topology; is complete.
• Uniform Cauchy criterion for uniform convergence Weierstrass M-test for series: if and converges then converges uniformly.
• Power series: . Radius of convergence: the series converges for , uniformly on for all , and diverges for . is continuous on (… and differentiable to all orders, see below).

3. Derivatives in 1 real variable

(Rudin ch. 5; McMullen §5; Lec. 20)

• Differentiable continuous.
• Mean value theorem: differentiable such that .
• Taylor’s theorem: times differentiable such that
• Most functions cannot be expressed as power series (the Taylor series ).
• , pointwise, uniformly , , and in the topology.
• with is a complete metric space.
• a power series is on , and .

4. Riemann integral

(Rudin ch. 6; McMullen §6; Lec. 21)

• bounded, , . Lower/upper Riemann sums: is Riemann integrable on if .

• ; ; etc.

• (piecewise) continuous on integrable.

• If then is differentiable and (fundamental theorem of calculus).

• if uniformly then .

• The norm: for , coarser than the uniform topology ( uniformly on in , but not ).

  inner product: .