Download TOPOLOGY, DR. BLOCK, SPRING 2016, NOTES, PART 6

TOPOLOGY, DR. BLOCK, SPRING 2016, NOTES, PART 6
600. Definition. Let f and g be continuous maps from a space X to a space Y. We
say that f is homotopic to g if and only if there is a continuous map F : X × I → Y
such that for all x ∈ X, we have F (x, 0) = f (x) and F (x, 1) = g(x). The map F is
called a homotopy between f and g. If f is homotopic to a constant map, we say
that f is nulhomotopic.
601. Definition. Let y and z be a fixed elements of a topological space X. Let p
and q be paths with initial point y and terminal point z. We say that p and q are
path homotopic if and only if there is a continuous function F : I × I → X such
that
1. For each s ∈ I, we have F(s,0) = p(s) and F(s,1) = q(s).
2. For each t ∈ I, F (0, t) = y and F (1, t) = z. The function F is called a path
homotopy between p and q.
602. Proposition. Let y and z be a fixed elements of a topological space X. Let p
and q be paths with initial point y and terminal point z. Let F be a path homotopy
between p and q. For each t ∈ I, define a function Ft : I → X by Ft (s) = F (s, t).
Then each Ft is a path with initial point y and terminal point z.
603. Theorem. Let X and Y be topological spaces. The relation ”homotopic”
is an equivalence relation on the set of all continuous maps from X to Y.
604. Theorem. Let y and z be a fixed elements of a topological space X. The
relation ”path homotopic” is an equivalence relation on the set of all paths with
initial point y and terminal point z.
605. Proposition. Let S be a convex subset of euclidean space Rn . Let y, z ∈ S,
and let p and q be paths in S with initial point y and terminal point z. Then p and
q are path homotopic.
606. Lemma. Suppose f : X → Y is continuous. Let y, z ∈ X, and let p and q
be paths in X with initial point y and terminal point z. Suppose that p and q are
path homotopic. Then the paths f ◦ p and f ◦ q in Y with initial point f (y) and
terminal point f (z) are also path homotopic.
607. Theorem. Suppose that X is a topological space, and f, g : X → R are
continuous. Then f + g, f − g, f · g are also continuous. Also, if g(x) 6= 0 for all
x ∈ X, then fg is continuous.
TOPOLOGY, DR. BLOCK, SPRING 2016, NOTES, PART 6
608. Lemma. Let w, x, y, z be fixed elements of a topological space X. Let p be
a path in X with initial point w and terminal point x. Let q be a path in X with
initial point x and terminal point y. Let r be a path in X with initial point y and
terminal point z. Then the paths (p ∗ q) ∗ r and p ∗ (q ∗ r) (which have initial point
w and terminal point z) are path homotopic.
609. Proposition. Let f be a path in X, and let 0 = a0 < a1 · · · < an = 1 be a
partition of [0, 1]. For each i = 1, . . . , n, let Li denote the unique linear map with
positive slope which maps I onto the interval [ai−1 , ai ], and let fi : I → X be the
composition fi = f ◦Li . Then the paths f and (f1 ∗f2 ∗· · ·∗fn ) are path homotopic.
610. Definition. Let z be an element of a topological space X. A path in X with
initial point z and terminal point z is called a loop with base point z. The constant
loop with base point z is the function c : I → X defined by c(t) = z for each t ∈ I.
611. Lemma. Let p be a path in X with initial point y and terminal point z,
and let c be the constant loop with base point z. Then the paths p and p ∗ c are
path homotopic.
612. Lemma. Let p be a path in X with initial point y and terminal point z,
and let c be the constant loop with base point y. Then the paths p and c ∗ p are
path homotopic.
613. Lemma. Let p be a path in X with initial point y and terminal point z,
and suppose that q is the reverse path of p. Let c be the constant loop with base
point y. Then the loops p ∗ q and c are path homotopic.
614. Lemma. Let p and r be paths in X with initial point x and terminal point
y which are path homotopic. Let q and w be paths in X with initial point y and
terminal point z which are path homotopic. Then p∗q and r∗w are path homotopic.
615. Definition. Let z be a fixed element of a topological space X. For a loop p
with base point z, let [p] denote the equivalence class of p, for the relation ”path
homotopic” . Let π1 (X, z) denote the set of equivalence classes. For elements [p]
and [q] of π1 (X, z) set [p]∗[q] = [p∗q]. (This is well defined by the previous Lemma.)
616. Theorem. π1 (X, z) with the operation ∗ is a group. (This group is called
the fundamental group of X at z. )
617. Theorem. Let X be a path connected space, and let y and z be points of
X. Then the groups π1 (X, y) and π1 (X, z) are isomorphic.
618. Problem. Suppose that f, g : X → Y are homotopic and k, h : Y → Z are
homotopic. Prove that k ◦ f and h ◦ g are homotopic.
TOPOLOGY, DR. BLOCK, SPRING 2016, NOTES, PART 6
619. Problem. Suppose that X is a space and the identity map of X to X is
nulhomotopic. Prove that X is path connected.
620. Lemma. Let X and Y be spaces with base points x0 and y0 respectively.
Let f : X → Y be continuous with f (x0 ) = y0 . Let fˆ : π1 (X, x0 ) → π1 (Y, y0 ) be
defined by fˆ([p]) = [f ◦ p]. Then fˆ is well defined and is a group homomorphism.
621. Theorem. Let X and Y be path connected spaces. If X and Y are homeomorphic, then their fundamental groups are isomorphic.
622. Definition. Let p : E → B be a continuous surjective map. Let W
be an open subset of B. We say that W is evenly covered by p if and only if
there exists a pairwise
disjoint collection {Vα : α ∈ A} of open subsets of E such
S
−1
that p (W ) = α∈A Vα and for each α ∈ A, the restriction of p to Vα is a
homomorphism onto W. The collection {Vα : α ∈ A} is called a partition of p−1 (W )
into slices.
623. Definition. Let p : E → B be a continuous surjective map. We say that p
is a covering map and E is a covering space of B if and only if every point x ∈ B
has a neighborhood W that is evenly covered by p.
624. Definition. Let S 1 denote {(x, y) ∈ R × R : x2 + y 2 = 1}.
625. Theorem. Let p : R → S 1 be defined by p(t) = (cos 2πt, sin 2πt). Then p is
a covering map.
626. Proposition. A covering map is an open map.
627. Example. Let n be a positive integer, and let p : S 1 → S 1 be defined by
p(z) = z n . Then p is a covering map.
628. Problem. Let p : E → B be a continuous surjective map. Let W be an
open subset of B that is evenly covered by p. Show that if W is connected then
the partition of p−1 (W ) into slices is unique. Give an example where W is not
connected and there are two distinct partitions of p−1 (W ) into slices.
629. Theorem. Let p : E → B and q : G → D be covering maps. Then the map
p×q :E×G→B×D
is a covering map.
630. Definition. Let p : E → B be a map. Let f : X → B be continuous. A
lifting of f is a continuous map g : X → E such that p ◦ g = f.
TOPOLOGY, DR. BLOCK, SPRING 2016, NOTES, PART 6
631. Theorem. Let p : E → B be a covering map. Let f : [0, 1] → B be a path
with initial point b0 . Let e0 be a point in E with p(e0 ) = b0 . Then there is a unique
path g in E, with initial point e0 , which is a lifting of f.
632. Theorem. Let p : E → B be a covering map. Let e0 be a point in E, and
let p(e0 ) = b0 . Let F : I × I → B be continuous with F (0, 0) = b0 . Then there is a
unique lifting of F to a continuous map G : I × I → E such that G(0, 0) = e0 . If F
is a path homotopy, then G is a path homotopy.
633. Theorem. Let p : E → B be a covering map. Let e0 be a point in E, and
let p(e0 ) = b0 . Let b1 ∈ B. Let f and g be two paths in B with initial point b0 and
terminal point b1 . Let f˜ and g̃ be the liftings of f and g to paths with initial point
e0 . If f and g are path homotopic, then f˜ and g̃ have the same terminal point and
are path homotopic.
634. Definition. Let p : E → B be a covering map. Let e0 be a point in E, and
let p(e0 ) = b0 . We define a function φ from the set π1 (B, b0 ) to the set p−1 (b0 ) as
follows. Let [f ] ∈ π1 (B, b0 ). Let f˜ be the lifting of f to a path with initial point e0 .
Let φ([f ]) = f˜(1). Note that φ is well defined by the previous theorem. We call φ
the lifting correspondence.
635. Definition. A space X is simply connected if and only if X is path connected
and the fundamental group of X is the group consisting only of an identity element.
636. Lemma. If X is simply connected, then any two paths in X which have the
same initial point and the same terminal point are path homotopic.
637. Theorem. Let p : E → B be a covering map. Let e0 be a point in E, and let
p(e0 ) = b0 . If E is path connected, then the lifting correspondence φ is surjective.
If E is simply connected, then the lifting correspondence φ is bijective.
638. Theorem. The fundamental group of S 1 is isomorphic to the group of
integers under addition.
639. Problem. Let p : E → B be a covering map. Suppose that E is path
connected and B is simply connected. Prove that p is a homeomorphism.
640. Definition. Let E ⊂ X. A retraction of X onto E is a continuous map
r : X → E such that r(y) = y for each y ∈ E. If there exists a continuous map
r : X → E such that r(y) = y for each y ∈ E, we say that E is a retract of X.
641. Problem. Suppose that h : X → Y is a homeomorphism. Prove that if E
is a retract of X, then h(E) is a retract of Y.
642. Problem. Suppose X is a Hausdorff space, and E is a retract of X. Prove
that E is a closed subset of X.
TOPOLOGY, DR. BLOCK, SPRING 2016, NOTES, PART 6
643. Definition. Let D2 denote {(x, y) ∈ R × R : x2 + y 2 ≤ 1}.
644. Proposition. S 1 is a retract of D2 − {(0, 0)}.
645. Proposition. Let E be a retract of X. Let i : E → X be defined by i(y) = y.
Let z ∈ E, and let î : π1 (E, z) → π1 (X, z) be the induced homomorphism of the
fundamental groups (as in 620). Then î is injective.
646. Theorem. S 1 is not a retract of D2 .
647. Lemma. The map p : S 1 ×I → D2 defined by p((x, y), t) = ((1−t)x, (1−t)y)
is a quotient map.
648. Theorem. Let X be a topological space, and let g : S 1 → X be continuous.
Suppose that g is nulhomotopic. Then there is a continuous extension f : D2 → X
of g.
649. Theorem. Let X be a topological space, and let g : S 1 → X be continuous.
Suppose there is a continuous extension f : D2 → X of g. Then the homomorphism
ĝ (induced by g) from the fundamental group of S 1 to the fundamental group of X
is the trivial homomorphism.
650. Theorem. Let X be a topological space, and let g : S 1 → X be continuous.
Suppose that the homomorphism ĝ (induced by g) from the fundamental group
of S 1 to the fundamental group of X is the trivial homomorphism. Then g is
nulhomotopic.
651. Corollary. The inclusion map j : S 1 → R2 − {(0, 0)} is not nulhomotopic.
The identity map i : S 1 → S 1 is not nulhomotopic.
652. Definition. Let E ⊂ R2 . A vector field on E is a continuous map v : E → R2 .
We think of v(x) as a vector with tail at x. The vector field v is nonvanishing if
and only if v : E → R2 − {(0, 0)}.
Suppose that S 1 ⊂ E. We say that the vector field v points directly outward
at x ∈ S 1 if and only if v(x) is a positive scalar multiple of x. We say that the
vector field v points directly inward at x ∈ S 1 if and only if v(x) is a negative scalar
multiple of x.
653. Theorem. Let v be a nonvanishing vector field on D2 . There exists x ∈ S 1
such that v points directly outward at x, and there exists y ∈ S 1 such that v points
directly inward at y.
654. Theorem (Brouwer fixed point theorem for the disc). Let f : D2 → D2 be
continuous. Then f has a fixed point. (There is a point x ∈ D2 with f (x) = x. )
TOPOLOGY, DR. BLOCK, SPRING 2016, NOTES, PART 6
655. Problem. Prove that if f : S 1 → S 1 is nulhomotopic, then f has a fixed
point and f maps some point x to its antipode −x.
656. Theorem. Let A be a 3 by 3 matrix of positive real numbers. Then A has
a positive real eigenvalue.
657. Problem. Prove that if f : S 1 → S 1 is given by f (z) = z n , where n is a
positive integer, then the map induced by f on the fundamental group is injective.
658. Theorem. (Fundamental theorem of Algebra). A polynomial equation of
the form
xn + an−1 xn−1 + · · · + a1 x + a0 = 0
where n ≥ 1 and the coefficients may be real or complex, has at least one (real or
complex) root.
659. Definition. Let S n denote
{(x1 , x2 , . . . , xn+1 ) ∈ Rn+1 : (x1 )2 + (x2 )2 + · · · + (xn+1 )2 = 1}.
Let p = (0, 0, . . . , 0, 1) ∈ Rn+1 and q = (0, 0, . . . , 0, −1) ∈ Rn+1 denote the ”north
pole” and ”south pole” respectively.
660. Definition. If x ∈ S n , then its antipode is the point −x. A map h : S n → S m
is antipode preserving if and only if h(−x) = −h(x) for all x ∈ S n .
661. Theorem. If h : S 1 → S 1 is continuous and antipode preserving, then h is
not nulhomotopic.
662. Theorem. There does not exist a continuous antipode preserving map
g : S2 → S1.
663. Theorem. (Borsuk-Ulam theorem for S 2 ) If f : S 2 → R2 is continuous,
then there is a point x ∈ S 2 with f (x) = f (−x).
664. Proposition. Define f : S n − {p} → Rn by
f (x1 , x2 , . . . , xn+1 ) =
1
(x1 , x2 , . . . , xn )
1 − xn+1
Then f is a homeomorphism. (This homeomorphism is called stereographic projection.)
665. Proposition. S n − {p} is homeomorphic to S n − {q}.
666. Definition. A topological space X is said to be homogeneous if and only if
for every pair of points x, y ∈ X there exists a homeomorphism h : X → X such
that h(x) = y.
TOPOLOGY, DR. BLOCK, SPRING 2016, NOTES, PART 6
667. Proposition. For any positive integer n the space S n is homogeneous.
668. Corollary. For any pair of points x, y ∈ S n the spaces S n −{x} and S n −{y}
are homeomorphic.
669. Problem. Prove that if g : S 2 → S 2 is continuous and g(x) 6= g(−x) for all
x ∈ S 2 , then g is surjective.
670. Definition. We use the notation f : (X, x0 ) → (Y, y0 ) to say that f : X → Y
and f (x0 ) = y0 .
671. Lemma. Suppose that h, k : (X, x0 ) → (Y, y0 ) are continuous maps. Suppose that h and k are homotopic, and there is a homotopy H : X × I → Y such
that H(x0 , t) = y0 for all t. Then the homomorphisms induced by h and k on the
fundamental group are equal.
672. Definition. Let A be a subspace of X. We say that A is a deformation
retract of X if and only if there is a continuous map H : X × I → X such that for
all x ∈ X, t ∈ I, a ∈ A we have
H(x, 0) = x, H(x, 1) ∈ A, H(a, t) = a.
The map H is called a deformation retraction.
673. Remark. If A is a deformation retract of X and H is a deformation
retraction, then the map r : X → A given by r(x) = H(x, 1) is a retraction of X
onto A. Moreover, if j : A → X is the inclusion, then H is a homotopy between the
identity map of X and j ◦ r.
674. Theorem. Suppose that A is a deformation retract of X, and let x0 ∈
A. Then the inclusion map j : (A, x0 ) → (X, x0 ) induces an isomoprhism of the
fundamental groups.
675. Problem. Suppose that A is a deformation retract of X, and B is a deformation retract A. Prove that that B is a deformation retract X.
676. Problem. Let n positive integer.
Prove that the spaces S n and Rn+1 − {0} have isomorphic fundamental groups.
677. Definition. Let X be a space, and let y and z be points of X. Suppose that
α is a path in X from y to z and β is the reverse path. We define α̂ : π1 (X, y) →
π1 (X, z) by α̂([p]) = [β ∗ p ∗ α].
678. Remark. We saw in Theorem 617 that α̂ is a group isomorphism.
TOPOLOGY, DR. BLOCK, SPRING 2016, NOTES, PART 6
679. Lemma. Let h, k : X → Y be continuous maps with h(x0 ) = y0 and
k(x0 ) = y1 . Suppose that h and k are homotopic. Let ĥ : π1 (X, x0 ) → π1 (Y, y0 )
and k̂ : π1 (X, x0 ) → π1 (Y, y1 ) be the induced maps on the fundamental groups.
Then there is a path α in Y from y0 to y1 such that k̂ = α̂ ◦ ĥ.
680. Corollary. Let h, k : X → Y be continuous maps with h(x0 ) = y0 and
k(x0 ) = y1 . Suppose that h and k are homotopic. Let ĥ : π1 (X, x0 ) → π1 (Y, y0 )
and k̂ : π1 (X, x0 ) → π1 (Y, y1 ) be the induced maps on the fundamental groups.
If ĥ is injective, then k̂ is injective. If ĥ is surjective, then k̂ is surjective. If ĥ is
trivial, then k̂ is trivial.
681. Corollary. Let h : X → Y be continuous and nulhomotopic. Then h induces
the trivial homomorphism on the fundamental group.
682. Definition. Let f : X → Y and g : Y → X be continuous maps. Suppose
that g ◦ f is homotopic to the identity map of X, and f ◦ g is homotopic to the
identity map of Y. Then the maps f and g are called homotopy equivalences, and
each map is called a homotopy inverse of the other. Also, we say that X and Y
have the same homotopy type.
683. Problem. Given any collection of topological spaces, prove that the relation
”have the same homotopy type” is an equivalence relation on the collection.
684. Remark. If A is a deformation retract of X, then A and X have the same
homotopy type.
685. Theorem. Let f : X → Y and be a continuous map with f (x0 ) = y0 . If
f is a homotopy equivalence, then the induced homorphism from the fundamental
group π1 (X, x0 ) to the fundamental group π1 (Y, y0 ) is an isomorphism.
686. Definition. A space X is said to be contractible if and only if the identity
map of X is nulhomotopic.
687. Problem. Prove that a space X is contractible if and only X has the same
homotopy type as a one point space.
688. Problem. Prove that a retract of a contractible space is contractible.
689. Problem. The ”figure eight” space is the union of two circles having a
point is common. The ”theta” space is the union of S 1 and [−1, 1] × {0}. Are the
fundamental groups of these two spaces isomorphic?
TOPOLOGY, DR. BLOCK, SPRING 2016, NOTES, PART 6
690. Definition. Let G be a group with group operation ∗. Let H be a subset
of G. We say that H generates G (or the elements of H generate G) if and only if
every element of g of G can be expressed as
g = h1 ∗ · · · ∗ hn
where for each i either hi ∈ H or hi is the inverse of an element of H.
691. Theorem. Suppose that X = V ∪ W, where V and W are open subsets of
X. Let x0 ∈ V ∩ W, and suppose that V ∩ W is path connected. Let i : V → X and
j : W → X. be the inclusion maps. Then the images of the induced homomorphisms
î : π1 (V, x0 ) → π1 (X, x0 ) and ĵ : π1 (W, x0 ) → π1 (X, x0 ) generate π1 (X, x0 ).
692. Remark. The previous theorem is a special case of a theorem called the
Seifert-van Kampen theorem.
693. Corollary. Suppose that X = V ∪ W, where V and W are open subsets of
X. Suppose that V ∩ W is nonempty and path connected. If V and W are simply
connected, then X is simply connected.
694. Theorem. If n ≥ 2, then S n is simply connected.
695. Definition and Proposition. Suppose G and H are groups with group
operations ∗G and ∗H . Then G × H is a group with group operation ∗ given by
(a, b) ∗ (c, d) = (a ∗G c, b ∗H d).
696. Theorem. Suppose that X and Y are spaces with x0 ∈ X and y0 ∈ Y. Then
π1 (X × Y, (x0 , y0 )) is isomorphic to π1 (X, x0 ) × π1 (Y, y0 ).