Topological Complexity and LS-Category of Certain Manifolds

Te Lusternik–Schnirelmann category and topological complexity are important invariants of topological spaces. In this paper, we calculate the Lusternik–Schnirelmann category and topological complexity of products of real projective spaces and their wedge products by using cup and zero-cup length. Also, we will fnd the topological complexity of R P 2 k + 1 by using the immersion dimension of R P 2 k + 1 .


Introduction
Te Lusternik-Schnirelmann category, (LS-category), of a topological space X which introduced in 1920, is an invariant of a manifold which gave a lower bound for the number of critical points of a function on a closed manifold.Te topological complexity is a numerical homotopy invariant, introduced by M. Farber in 2001.M. Farber examined the topological complexity of the robotics [1][2][3].Topological complexity has close relationship to classical invariant Lusternik-Schnirelmann category.
Defnition 1 (see [4]).Te Lusternik-Schnirelmann category of a space X is the least integer n such that there exists an open covering U 1 , . . ., U n+1 of X with each U i contractible to a point in the space X.We denote this by cat(X) � n and we call such a covering U i   categorical.If no such integer exists, we write cat(X) � ∞.
Defnition 2 (see [1]).Let π: PX ⟶ X × X be the path fbration.Topological complexity of a topological space X, denoted by TC(X), is the least nonnegative integer k if there are open subsets U 0 , U 1 , . . ., U k which cover X × X such that on each U i there exists a continuous section of π for i � 0, 1, . . ., k. Tis paper is organized as follows.In Section 2, we will calculate Lusternik-Schnirelmann category of products of real projective spaces utilizing [5,6].In Section 3, we will calculate topological complexity of products of real projective spaces by using the results of [7].Furthermore, the topological complexity of wedge products of real projective spaces is calculated by using the results of Section 2. Section 4 provides the topological complexity calculation of RP 2 k +1 by using [8,9] and formulates general results from previous sections.Additionally, general examples are given.
Troughout this paper, we denote the immersion dimension of X by imd(X).

LS-Category of the Products of the Real Projective Spaces
Tis section is devoted to calculating LS-category of the products of real projective spaces by using cup-length.
Defnition 3 (see [4]).Let R be a commutative ring and X be a space.Te cup-length of X with coefcients in R is the least integer k(or ∞) such that all (k + 1)-fold cup products vanish in the reduced cohomology  H * (X; R); we denote this integer by cup R (X).
To prove the main theorem of this section, we use the following results of [4].Proposition 4 (see [4]).Te R-cuplength of a space is less than or equal to the category of the space for all coefcients R. In notation, we write cup R (X) ≤ cat(X).

Topological Complexity of Products and Wedge Products of Real Projective Spaces
In this section, we will calculate the topological complexity of the products and wedge products of real projective spaces.We also give a lower bound for TC(X), where X is the product of real projective spaces.Te lower bound is quite useful since it allows an efective computation of TC(X) in many examples.A lower bound for topological complexity is also obtained by using the zero-divisor-cup-length of X.
Defnition 10 (see [1]).Let K be a feld.Te kernel of homomorphism ∪ : H * (X; K) ⊗ H * (X; K) ⟶ H * (X; K) is called the ideal of the zero-divisors of H * (X; K).Te zerodivisors-cup-length of H * (X; K) is the length of the longest nontrivial product in the ideal of the zero-divisors of H * (X; K).Tis number will be denoted by zcl(X).
Lemma 15.For any positive integers k i for i � 1, 2, . . ., n, we have be defned by We may show by easy calculation that α , i s are in the kernel of Clearly, and calculation reveals that Similarly, we can show that α □ Lemma 1 .For any positive integers k i , i � 1, 2, . . ., n, we have .
Proof.By Teorems 12 and 13, Journal of Mathematics Clearly, Lemmas 15 and 16 imply the following result.

□
Theorem 17.For any positive integers k i , i � 1, 2, . . ., n, we have Corollary 18.For any positive integer k, we have Remark 19.By Teorem 17, clearly Teorem 7.1 in [10] is true for Note that we do not know if this is true for arbitrary products.
To calculate topological complexity of the wedge products of real projective spaces, we use the next theorem from [11].
Theorem 20 (see [11]).Let X, Y be Hausdorf normal topological spaces and path connected with nondegenerate basepoints, such that X × X, Y × Y and X × Y are normal.Ten, Theorem 21.For any positive integers, Proof.Proof follows by induction on k.If k � 2, then by Teorem 20, we have We recall that cat(X∨Y) � max cat(X), cat(Y) { }, so by induction, we have

More Examples on Topological Complexity
First, we calculate topological complexity of P 2 k +1 by immersion dimension of P 2 k +1 .
Using Remark 14 and Lemma 15 gives us the following lemma.
From Example 1, we may fll nonimmersion and immersion parts for m � 2 k + 1 in the Don Davis table of immersion and embedding of real projective spaces.
By calculating the zero cup-length of product and Teorem 12, we have the next proposition.

Proposition 2 . Let
Proof.Te proof follows by calculating the zero cup-length of product and Teorem 12.
Next example shows that Proposition 26 is not true for any arbitrary product of real projective spaces.Consider the following examples.
Example 3. Note that TC(P 10 ) � imd(P As we see there is a gap between lower and upper bounds.
For 2 k + 1 < n < 2 k+1 in Example 3, to calculate topological complexity of P n × P n from zero cup-length that we have used in our calculation, we will fnd a gap between lower and upper bounds of topological complexity.Terefore, we cannot use this technique to fnd topological complexity of arbitrary products.